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Research ArticleLinear Approximation and Asymptotic Expansion ofSolutions for a Nonlinear Carrier Wave Equation in an AnnularMembrane with Robin-Dirichlet Conditions
Le Thi Phuong Ngoc1 Le Huu Ky Son23 TranMinh Thuyet4 and Nguyen Thanh Long3
1University of Khanh Hoa 01 Nguyen Chanh Str Nha Trang City Vietnam2Department of Fundamental sciences Ho Chi Minh City University of Food Industry 140 Le Trong Tan StrTan Phu Dist Ho Chi Minh City Vietnam3Department of Mathematics and Computer Science University of Natural ScienceVietnam National University-Ho Chi Minh City 227 Nguyen Van Cu Str Dist 5 Ho Chi Minh City Vietnam4Department of Mathematics University of Economics of Ho Chi Minh City59C Nguyen Dinh Chieu Str Dist 3 Ho Chi Minh City Vietnam
Correspondence should be addressed to NguyenThanh Long longnt2gmailcom
Received 23 April 2016 Accepted 8 September 2016
Academic Editor Yuri Vladimirovich Mikhlin
Copyright copy 2016 LeThi Phuong Ngoc et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
This paper is devoted to the study of a nonlinear Carrier wave equation in an annular membrane associated with Robin-Dirichletconditions Existence and uniqueness of aweak solution are proved by using the linearizationmethod for nonlinear terms combinedwith the Faedo-Galerkin method and the weak compact method Furthermore an asymptotic expansion of a weak solution of highorder in a small parameter is established
1 Introduction
In this paper we consider the following nonlinear Carrierwave equation in the annular membrane
where120583119891 0 1 are given functions 120588 120577 are given constantswith 0 lt 120588 lt 1 In (1) nonlinear term 120583(119906(119905)20) depends onintegral 119906(119905)20 = int1
1205881199091199062(119909 119905)119889119909
Equation (1) herein is the bidimensional nonlinear waveequation describing nonlinear vibrations of annular mem-brane Ω1 = (119909 119910) 1205882 lt 1199092 + 1199102 lt 1 In the vibration pro-cessing the area of the annular membrane and the tensionat various points change in time The condition on boundaryΓ1 = (119909 119910) 1199092 + 1199102 = 1 that is 119906119909(1 119905) + 120577119906(1 119905) = 0describes elastic constraints where 120577 constant has a mechan-ical signification And with the boundary condition on Γ120588 =(119909 119910) 1199092+1199102 = 1205882 requiring 119906(120588 119905) = 0 the annularmem-brane is fixed
In [1] Carrier established the equation which modelsvibrations of an elastic string when changes in tension are notsmall
120588119906119905119905 minus (1 + 1198641198601198711198790 int119871
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8031638 18 pageshttpdxdoiorg10115520168031638
2 Mathematical Problems in Engineering
where 119906(119909 119905) is 119909-derivative of the deformation 1198790 is thetension in the rest position 119864 is the Young modulus 119860 is thecross section of a string 119871 is the length of a string and 120588 is thedensity of a material Clearly if properties of a material varywith 119909 and 119905 then there is a hyperbolic equation of the type[2]
119906119905119905 minus 119861(119909 119905 int1
The Kirchhoff-Carrier equations of form (1) receivedmuch attention We refer the reader to for example Caval-canti et al [3 4] Ebihara et al [5] Miranda and Jutuca [6]Lasiecka and Ong [7] Hosoya and Yamada [8] Larkin [2]Medeiros [9] Menzala [10] Park et al [11 12] Rabello et al[13] and Santos et al [14] for many interesting results andfurther references
Thepaper consists of four sections Preliminaries are donein Section 2 with the notations definitions list of appropriatespaces and required lemmas The main results are presentedin Sections 3 and 4
First by combining the linearization method for nonlin-ear terms the Faedo-Galerkinmethod and theweak compactmethod we prove that problem (1)ndash(3) has a unique weaksolution
Next by using Taylorrsquos expansion of given functions 120583 1205831119891 and 1198911 up to high order119873 + 1 we establish an asymptoticexpansion of solution 119906 = 119906120576 of order119873+1 in small parameter120576 for
120588 lt 119909 lt 1 0 lt 119905 lt 119879 associated with (1) and (2) with 120583 isin119862119873+1(R+) 1205831 isin 119862119873(R+) 120583(119911) ge 120583lowast gt 0 1205831(119911) ge 0 for all119911 isin R+119891 isin 119862119873+1([120588 1]timesR+timesR3)1198911 isin 119862119873([120588 1]timesR+timesR3)Our results can be regarded as an extension and improvementof the corresponding results of [15 16]
2 Preliminaries
First put Ω = (120588 1) 119876119879 = Ω times (0 119879) 119879 gt 0 We omit thedefinitions of the usual function spaces and denote them bynotations 119871119901 = 119871119901(Ω) 119867119898 = 119867119898(Ω) Let (sdot sdot) be a scalarproduct in 1198712 Notation sdot stands for the norm in 1198712 and wedenote sdot 119883 the norm in Banach space119883 We call1198831015840 the dualspace of 119883 We denote 119871119901(0 119879119883) 1 le 119901 le infin the Banachspace of real functions 119906 (0 119879) rarr 119883 to be measurable suchthat 119906119871119901(0119879119883) lt +infin with
119906119871119901(0119879119883) = (int119879
0119906 (119905)119901119883 119889119905)1119901 if 1 le 119901 lt infin
ess sup0lt119905lt119879
119906 (119905)119883 if 119901 = infin (7)
With 119891 isin 119862119896([120588 1] times R+ times R3) 119891 = 119891(119909 119905 1199101 1199102 1199103)we put 1198631119891 = 120597119891120597119909 1198632119891 = 120597119891120597119905 119863119894+2119891 = 120597119891120597119910119894 with
The norms in 11987121198671 and1198672 induced by the correspond-ing scalar products (10) are denoted by sdot 0 sdot 1 and sdot 2
Consider the following set119881 = V isin 1198671 V (120588) = 0 (11)
It is obviously that 119881 is a closed subspace of 1198671 and on119881 two norms V1198671 and V119909 are equivalent norms On theother hand 119881 is continuously and densely embedded in 1198712Identifying 1198712 with (1198712)1015840 (the dual of1198712) we have119881 997893rarr 1198712 997893rarr1198811015840 We note more that the notation ⟨sdot sdot⟩ is also used for thepairing between 119881 and 1198811015840
We then have the following lemmas
Lemma 1 The following inequalities are fulfilled
(i) radic120588V le V0 le V for all V isin 1198712(ii) radic120588V1198671 le V1 le V1198671 for all V isin 1198671
Proof of Lemma 1 It is easy to verify the above inequalities viathe following inequalities
120588int1
120588V2 (119909) 119889119909 le int1
120588119909V2 (119909) 119889119909 le int1
120588V2 (119909) 119889119909
forallV isin 1198712120588 int1
120588V2119909 (119909) 119889119909 le int1
120588119909V2119909 (119909) 119889119909 le int1
Lemma 2 Embedding 119881 997893rarr 1198620(Ω) is compact and for all V isin119881 we have(i) V1198620(Ω) le radic1 minus 120588V119909(ii) V le ((1 minus 120588)radic2)V119909(iii) V0 le ((1 minus 120588)radic2120588)V1199090(iv) V11990920 + V2(1) ge V20(v) |V(1)| le radic3V1
Mathematical Problems in Engineering 3
Proof of Lemma 2 Embedding 119881 997893rarr 1198671 is continuous andembedding 1198671 997893rarr 1198620(Ω) is compact so embedding 119881 997893rarr1198620(Ω) is compact In what follows we prove (i)ndash(v)
(i) For all V isin 119881 and 119909 isin [120588 1]|V (119909)| = 100381610038161003816100381610038161003816100381610038161003816int119909
120588V119909 (119910) 119889119910100381610038161003816100381610038161003816100381610038161003816 le int1
120588
1003816100381610038161003816V119909 (119910)1003816100381610038161003816 119889119910le radic1 minus 120588 1003817100381710038171003817V1199091003817100381710038171003817 (13)
(ii) For all V isin 119881and 119909 isin [120588 1]V2 (119909) = 100381610038161003816100381610038161003816100381610038161003816int119909
120588V119909 (119910) 1198891199101003816100381610038161003816100381610038161003816100381610038162 le (119909 minus 120588)int119909
120588V2119909 (119910) 119889119910
le (119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 (14)
Integrating over 119909 from 120588 to 1 we obtainV2 = int1
120588V2 (119909) 119889119909 le int1
120588(119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 119889119909
= (1 minus 120588)22 1003817100381710038171003817V11990910038171003817100381710038172 (15)
(iii) For all V isin 119881V0 le V le 1 minus 120588radic2 1003817100381710038171003817V1199091003817100381710038171003817 le 1 minus 120588radic2120588 1003817100381710038171003817V11990910038171003817100381710038170 (16)
le 2 V20 + 2 V0 1003817100381710038171003817V11990910038171003817100381710038170 le 2 V20 + V20 + 1003817100381710038171003817V119909100381710038171003817100381720le 3 V21 (18)
implying (v)
Lemma 2 is proved
Remark 3 On 1198712 two norms V 997891rarr V and V 997891rarr V0 areequivalent So are two norms V 997891rarr V1198671 and V 997891rarr V1 on1198671and five norms V 997891rarr V1198671 V 997891rarr V1 V 997891rarr V119909 V 997891rarr V1199090and V 997891rarr radicV11990920 + V2(1) on 119881
where 120577 ge 0 is a constantLemma 4 Symmetric bilinear form 119886(sdot sdot) defined by (19) iscontinuous on 119881 times 119881 and coercive on 119881 that is
(i) |119886(119906 V)| le 11986211199061V1(ii) 119886(V V) ge 1198620V21
for all 119906 V isin 119881 where 1198620 = (12)min1 2120588(1 minus 120588)2 and1198621 = 1 + 3120577Proof of Lemma 4 (i) Byradic1 minus 120588V119909 ge V1198620(Ω) ge |V(1)| andradic120588V119909 le V1199090 for all V isin 119881 we have
ge 1003817100381710038171003817V119909100381710038171003817100381720 = 12 1003817100381710038171003817V119909100381710038171003817100381720 + 12 1003817100381710038171003817V119909100381710038171003817100381720ge 12 1003817100381710038171003817V119909100381710038171003817100381720 + 12 2120588(1 minus 120588)2 V20ge 12 min1 2120588(1 minus 120588)2V21
(21)
Lemma 4 is proved
Lemma 5 There exists Hilbert orthonormal base 119908119895 of space1198712 consisting of eigenfunctions119908119895 corresponding to eigenvalues120582119895 such that
(i) 0 lt 1205821 le 1205822 le sdot sdot sdot le 120582119895 le 120582119895+1 le sdot sdot sdot lim119895rarr+infin120582119895 =+infin(ii) 119886(119908119895 V) = 120582119895⟨119908119895 V⟩ for all V isin 119881 119895 = 1 2 Furthermore sequence 119908119895radic120582119895 is also the Hilbert
orthonormal base of 119881 with respect to scalar product 119886(sdot sdot)
4 Mathematical Problems in Engineering
On the other hand we also have119908119895 satisfying the followingboundary value problem
Proof The proof of Lemma 5 can be found in [17 p 87Theorem 77] with119867 = 1198712 and 119886(sdot sdot) as defined by (19)
We also note that operator119860 119881 rarr 1198811015840 in (22) is uniquelydefined by Lax-Milgramrsquos lemma that is119886 (119906 V) = ⟨119860119906 V⟩ forall119906 V isin 119881 (23)
Lemma 6 On 119881 cap 1198672 three norms V 997891rarr V1198672 V 997891rarr V2 =radicV20 + V11990920 + V11990911990920 and V 997891rarr V2lowast = radicV11990920 + 119860V20 areequivalent
Proof of Lemma 6 (i) It is easy to see that on 119881 cap 1198672 twonorms V 997891rarr V1198672 V 997891rarr V2 = radicV20 + V11990920 + V11990911990920 areequivalent becauseradic120588 V1198672 le V2 le V1198672 forallV isin 1198672 (24)
(ii) For all 119909 isin [120588 1] and V isin 119881 cap 1198672 we have
This implies1003817100381710038171003817119906119909119909100381710038171003817100381720 le 2 11986011990620 + 61205882 1003817100381710038171003817119906119909100381710038171003817100381720le 2(1 + 31205882) (11986011990620 + 1003817100381710038171003817119906119909100381710038171003817100381720)le 2(1 + 31205882) 11990622lowast
(28)
By V0 le ((1 minus 120588)radic2120588)V1199090 for all V isin 119881 we have11990622 = 11990620 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 minus 120588)22120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 + (1 minus 120588)22120588 ) 11990622lowast + 2(1 + 31205882) 11990622lowast= ((1 minus 120588)22120588 + 3 + 61205882)11990622lowast
(29)
(b) Proof 1199062lowast le const1199062It follows from (25) that
Remark 7 The weak formulation of initial-boundary valueproblem (1)ndash(3) can be given in the following manner find119906 isin 119882 = 119906 isin 119871infin(0 119879 119881 cap 1198672) 119906119905 isin 119871infin(0 119879 119881) 119906119905119905 isin119871infin(0 119879 1198712) such that 119906 satisfies the following variationalequation ⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (34)
Mathematical Problems in Engineering 5
for all V isin 119881 ae 119905 isin (0 119879) together with the initialconditions 119906 (0) = 0119906119905 (0) = 1 (35)
where 119886(sdot sdot) is the symmetric bilinear form on 119881 defined by(19)
3 The Existence and Uniqueness Theorem
Now we shall consider problem (1)ndash(3) with constant 120577 ge 0and make the following assumptions
(1198671) 0 isin 119881 cap 1198672 1 isin 119881(1198672) 120583 isin 1198621(R+) with 120583(119911) ge 120583lowast gt 0 forall119911 isin R+
(1198673) 119891 isin 1198620(Ω times R+ times R3) such that 119891(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+ timesR and119863119894119891 isin 1198620(ΩtimesR+ timesR3) 119894 =1 3 4 5Considering 119879lowast gt 0 fixed and letting 119879 isin (0 119879lowast] and119872 gt 0 we put119872 (120583) = sup
Theorem 8 Let assumptions (1198671)ndash(1198673) hold Then thereexist positive constants 119872 119879 such that the problem (40) (41)has solution 119906119898 isin 1198821(119872 119879)Proof of Theorem 8 It consists of three stepsStep 1 (the Faedo-Galerkin approximation (introduced byLions [18])) Consider basis 119908119895 for 119881 as in Lemma 5 Put
120572(119896)119895 119908119895 997888rarr 0 strongly in 119881 cap11986721199061119896 = 119896sum
119895=1
120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)
The system of (43) can be rewritten in form119888(119896)119898119895 (119905) + 120582119895120583119898 (119905) 119888(119896)119898119895 (119905) = 119865119898119895 (119905) 1 le 119895 le 119896119888(119896)119898119895 (0) = 120572(119896)119895 119888(119896)119898119895 (0) = 120573(119896)119895
(45)
in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)
6 Mathematical Problems in Engineering
Note that by (39) it is not difficult to prove that system(45) (46) has a unique solution 119888(119896)119898119895 (119905) 1 le 119895 le 119896 on interval[0 119879] so let us omit the details
Second Term 1198682 By the Cauchy-Schwartz inequality it gives100381610038161003816100381611986821003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905
By means of the convergences in (44) we can deduce theexistence of constant 119872 gt 0 independent of 119896 and 119898 suchthat 119878(119896)119898 (0) = 10038171003817100381710038171199061119896100381710038171003817100381720 + 119886 (1199061119896 1199061119896)+ 120583 (10038171003817100381710038170100381710038171003817100381720) [119886 (1199060119896 1199060119896) + 10038171003817100381710038171198601199060119896100381710038171003817100381720]
le 121198722(64)
for all119898 119896 isin NTherefore from (63) and (64) we can choose 119879 isin (0 119879lowast]
such that
(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)
Finally it follows from (62) (64) and (65) that119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872))+ 1198632 (119872)int119905
0119878(119896)119898 (119904) 119889119904 (67)
By using Gronwallrsquos Lemma (67) yields119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872)) exp (1199051198632 (119872)) le 1198722 (68)
for all 119905 isin [0 119879] for all119898 and 119896Therefore we have119906(119896)119898 isin 119882 (119872119879) forall119898 119896 (69)
Step 3 (limiting process) From (69) there exists a subse-quence of 119906(119896)119898 still so denoted such that
119906(119896)119898 997888rarr 119906119898 in 119871infin (0 119879 119881 cap 1198672) weaklylowast(119896)119898 997888rarr 1199061015840119898 in 119871infin (0 119879 119881) weaklylowast(119896)119898 997888rarr 11990610158401015840119898 in 1198712 (119876119879) weakly119906119898 isin 119882 (119872119879)
(70)
Passing to limit in (43) we have 119906119898 satisfying (40) (41) in1198712(0 119879) On the other hand it follows from (40)1 and (70)4that 11990610158401015840119898 = minus120583119898(119905)119860119906119898 + 119865119898 isin 119871infin(0 119879 1198712) and hence 119906119898 isin1198821(119872 119879) and the proof of Theorem 8 is complete
We will use the result obtained in Theorem 8 and thecompact imbedding theorems to prove the existence anduniqueness of a weak solution of problem (1)ndash(3) Hence weget the main result in this section
8 Mathematical Problems in Engineering
Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate
1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)
Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])
We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem
Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)
(76)
Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840
Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)
sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879)
(78)
Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
where 119906(119909 119905) is 119909-derivative of the deformation 1198790 is thetension in the rest position 119864 is the Young modulus 119860 is thecross section of a string 119871 is the length of a string and 120588 is thedensity of a material Clearly if properties of a material varywith 119909 and 119905 then there is a hyperbolic equation of the type[2]
119906119905119905 minus 119861(119909 119905 int1
The Kirchhoff-Carrier equations of form (1) receivedmuch attention We refer the reader to for example Caval-canti et al [3 4] Ebihara et al [5] Miranda and Jutuca [6]Lasiecka and Ong [7] Hosoya and Yamada [8] Larkin [2]Medeiros [9] Menzala [10] Park et al [11 12] Rabello et al[13] and Santos et al [14] for many interesting results andfurther references
Thepaper consists of four sections Preliminaries are donein Section 2 with the notations definitions list of appropriatespaces and required lemmas The main results are presentedin Sections 3 and 4
First by combining the linearization method for nonlin-ear terms the Faedo-Galerkinmethod and theweak compactmethod we prove that problem (1)ndash(3) has a unique weaksolution
Next by using Taylorrsquos expansion of given functions 120583 1205831119891 and 1198911 up to high order119873 + 1 we establish an asymptoticexpansion of solution 119906 = 119906120576 of order119873+1 in small parameter120576 for
120588 lt 119909 lt 1 0 lt 119905 lt 119879 associated with (1) and (2) with 120583 isin119862119873+1(R+) 1205831 isin 119862119873(R+) 120583(119911) ge 120583lowast gt 0 1205831(119911) ge 0 for all119911 isin R+119891 isin 119862119873+1([120588 1]timesR+timesR3)1198911 isin 119862119873([120588 1]timesR+timesR3)Our results can be regarded as an extension and improvementof the corresponding results of [15 16]
2 Preliminaries
First put Ω = (120588 1) 119876119879 = Ω times (0 119879) 119879 gt 0 We omit thedefinitions of the usual function spaces and denote them bynotations 119871119901 = 119871119901(Ω) 119867119898 = 119867119898(Ω) Let (sdot sdot) be a scalarproduct in 1198712 Notation sdot stands for the norm in 1198712 and wedenote sdot 119883 the norm in Banach space119883 We call1198831015840 the dualspace of 119883 We denote 119871119901(0 119879119883) 1 le 119901 le infin the Banachspace of real functions 119906 (0 119879) rarr 119883 to be measurable suchthat 119906119871119901(0119879119883) lt +infin with
119906119871119901(0119879119883) = (int119879
0119906 (119905)119901119883 119889119905)1119901 if 1 le 119901 lt infin
ess sup0lt119905lt119879
119906 (119905)119883 if 119901 = infin (7)
With 119891 isin 119862119896([120588 1] times R+ times R3) 119891 = 119891(119909 119905 1199101 1199102 1199103)we put 1198631119891 = 120597119891120597119909 1198632119891 = 120597119891120597119905 119863119894+2119891 = 120597119891120597119910119894 with
The norms in 11987121198671 and1198672 induced by the correspond-ing scalar products (10) are denoted by sdot 0 sdot 1 and sdot 2
Consider the following set119881 = V isin 1198671 V (120588) = 0 (11)
It is obviously that 119881 is a closed subspace of 1198671 and on119881 two norms V1198671 and V119909 are equivalent norms On theother hand 119881 is continuously and densely embedded in 1198712Identifying 1198712 with (1198712)1015840 (the dual of1198712) we have119881 997893rarr 1198712 997893rarr1198811015840 We note more that the notation ⟨sdot sdot⟩ is also used for thepairing between 119881 and 1198811015840
We then have the following lemmas
Lemma 1 The following inequalities are fulfilled
(i) radic120588V le V0 le V for all V isin 1198712(ii) radic120588V1198671 le V1 le V1198671 for all V isin 1198671
Proof of Lemma 1 It is easy to verify the above inequalities viathe following inequalities
120588int1
120588V2 (119909) 119889119909 le int1
120588119909V2 (119909) 119889119909 le int1
120588V2 (119909) 119889119909
forallV isin 1198712120588 int1
120588V2119909 (119909) 119889119909 le int1
120588119909V2119909 (119909) 119889119909 le int1
Lemma 2 Embedding 119881 997893rarr 1198620(Ω) is compact and for all V isin119881 we have(i) V1198620(Ω) le radic1 minus 120588V119909(ii) V le ((1 minus 120588)radic2)V119909(iii) V0 le ((1 minus 120588)radic2120588)V1199090(iv) V11990920 + V2(1) ge V20(v) |V(1)| le radic3V1
Mathematical Problems in Engineering 3
Proof of Lemma 2 Embedding 119881 997893rarr 1198671 is continuous andembedding 1198671 997893rarr 1198620(Ω) is compact so embedding 119881 997893rarr1198620(Ω) is compact In what follows we prove (i)ndash(v)
(i) For all V isin 119881 and 119909 isin [120588 1]|V (119909)| = 100381610038161003816100381610038161003816100381610038161003816int119909
120588V119909 (119910) 119889119910100381610038161003816100381610038161003816100381610038161003816 le int1
120588
1003816100381610038161003816V119909 (119910)1003816100381610038161003816 119889119910le radic1 minus 120588 1003817100381710038171003817V1199091003817100381710038171003817 (13)
(ii) For all V isin 119881and 119909 isin [120588 1]V2 (119909) = 100381610038161003816100381610038161003816100381610038161003816int119909
120588V119909 (119910) 1198891199101003816100381610038161003816100381610038161003816100381610038162 le (119909 minus 120588)int119909
120588V2119909 (119910) 119889119910
le (119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 (14)
Integrating over 119909 from 120588 to 1 we obtainV2 = int1
120588V2 (119909) 119889119909 le int1
120588(119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 119889119909
= (1 minus 120588)22 1003817100381710038171003817V11990910038171003817100381710038172 (15)
(iii) For all V isin 119881V0 le V le 1 minus 120588radic2 1003817100381710038171003817V1199091003817100381710038171003817 le 1 minus 120588radic2120588 1003817100381710038171003817V11990910038171003817100381710038170 (16)
le 2 V20 + 2 V0 1003817100381710038171003817V11990910038171003817100381710038170 le 2 V20 + V20 + 1003817100381710038171003817V119909100381710038171003817100381720le 3 V21 (18)
implying (v)
Lemma 2 is proved
Remark 3 On 1198712 two norms V 997891rarr V and V 997891rarr V0 areequivalent So are two norms V 997891rarr V1198671 and V 997891rarr V1 on1198671and five norms V 997891rarr V1198671 V 997891rarr V1 V 997891rarr V119909 V 997891rarr V1199090and V 997891rarr radicV11990920 + V2(1) on 119881
where 120577 ge 0 is a constantLemma 4 Symmetric bilinear form 119886(sdot sdot) defined by (19) iscontinuous on 119881 times 119881 and coercive on 119881 that is
(i) |119886(119906 V)| le 11986211199061V1(ii) 119886(V V) ge 1198620V21
for all 119906 V isin 119881 where 1198620 = (12)min1 2120588(1 minus 120588)2 and1198621 = 1 + 3120577Proof of Lemma 4 (i) Byradic1 minus 120588V119909 ge V1198620(Ω) ge |V(1)| andradic120588V119909 le V1199090 for all V isin 119881 we have
ge 1003817100381710038171003817V119909100381710038171003817100381720 = 12 1003817100381710038171003817V119909100381710038171003817100381720 + 12 1003817100381710038171003817V119909100381710038171003817100381720ge 12 1003817100381710038171003817V119909100381710038171003817100381720 + 12 2120588(1 minus 120588)2 V20ge 12 min1 2120588(1 minus 120588)2V21
(21)
Lemma 4 is proved
Lemma 5 There exists Hilbert orthonormal base 119908119895 of space1198712 consisting of eigenfunctions119908119895 corresponding to eigenvalues120582119895 such that
(i) 0 lt 1205821 le 1205822 le sdot sdot sdot le 120582119895 le 120582119895+1 le sdot sdot sdot lim119895rarr+infin120582119895 =+infin(ii) 119886(119908119895 V) = 120582119895⟨119908119895 V⟩ for all V isin 119881 119895 = 1 2 Furthermore sequence 119908119895radic120582119895 is also the Hilbert
orthonormal base of 119881 with respect to scalar product 119886(sdot sdot)
4 Mathematical Problems in Engineering
On the other hand we also have119908119895 satisfying the followingboundary value problem
Proof The proof of Lemma 5 can be found in [17 p 87Theorem 77] with119867 = 1198712 and 119886(sdot sdot) as defined by (19)
We also note that operator119860 119881 rarr 1198811015840 in (22) is uniquelydefined by Lax-Milgramrsquos lemma that is119886 (119906 V) = ⟨119860119906 V⟩ forall119906 V isin 119881 (23)
Lemma 6 On 119881 cap 1198672 three norms V 997891rarr V1198672 V 997891rarr V2 =radicV20 + V11990920 + V11990911990920 and V 997891rarr V2lowast = radicV11990920 + 119860V20 areequivalent
Proof of Lemma 6 (i) It is easy to see that on 119881 cap 1198672 twonorms V 997891rarr V1198672 V 997891rarr V2 = radicV20 + V11990920 + V11990911990920 areequivalent becauseradic120588 V1198672 le V2 le V1198672 forallV isin 1198672 (24)
(ii) For all 119909 isin [120588 1] and V isin 119881 cap 1198672 we have
This implies1003817100381710038171003817119906119909119909100381710038171003817100381720 le 2 11986011990620 + 61205882 1003817100381710038171003817119906119909100381710038171003817100381720le 2(1 + 31205882) (11986011990620 + 1003817100381710038171003817119906119909100381710038171003817100381720)le 2(1 + 31205882) 11990622lowast
(28)
By V0 le ((1 minus 120588)radic2120588)V1199090 for all V isin 119881 we have11990622 = 11990620 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 minus 120588)22120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 + (1 minus 120588)22120588 ) 11990622lowast + 2(1 + 31205882) 11990622lowast= ((1 minus 120588)22120588 + 3 + 61205882)11990622lowast
(29)
(b) Proof 1199062lowast le const1199062It follows from (25) that
Remark 7 The weak formulation of initial-boundary valueproblem (1)ndash(3) can be given in the following manner find119906 isin 119882 = 119906 isin 119871infin(0 119879 119881 cap 1198672) 119906119905 isin 119871infin(0 119879 119881) 119906119905119905 isin119871infin(0 119879 1198712) such that 119906 satisfies the following variationalequation ⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (34)
Mathematical Problems in Engineering 5
for all V isin 119881 ae 119905 isin (0 119879) together with the initialconditions 119906 (0) = 0119906119905 (0) = 1 (35)
where 119886(sdot sdot) is the symmetric bilinear form on 119881 defined by(19)
3 The Existence and Uniqueness Theorem
Now we shall consider problem (1)ndash(3) with constant 120577 ge 0and make the following assumptions
(1198671) 0 isin 119881 cap 1198672 1 isin 119881(1198672) 120583 isin 1198621(R+) with 120583(119911) ge 120583lowast gt 0 forall119911 isin R+
(1198673) 119891 isin 1198620(Ω times R+ times R3) such that 119891(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+ timesR and119863119894119891 isin 1198620(ΩtimesR+ timesR3) 119894 =1 3 4 5Considering 119879lowast gt 0 fixed and letting 119879 isin (0 119879lowast] and119872 gt 0 we put119872 (120583) = sup
Theorem 8 Let assumptions (1198671)ndash(1198673) hold Then thereexist positive constants 119872 119879 such that the problem (40) (41)has solution 119906119898 isin 1198821(119872 119879)Proof of Theorem 8 It consists of three stepsStep 1 (the Faedo-Galerkin approximation (introduced byLions [18])) Consider basis 119908119895 for 119881 as in Lemma 5 Put
120572(119896)119895 119908119895 997888rarr 0 strongly in 119881 cap11986721199061119896 = 119896sum
119895=1
120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)
The system of (43) can be rewritten in form119888(119896)119898119895 (119905) + 120582119895120583119898 (119905) 119888(119896)119898119895 (119905) = 119865119898119895 (119905) 1 le 119895 le 119896119888(119896)119898119895 (0) = 120572(119896)119895 119888(119896)119898119895 (0) = 120573(119896)119895
(45)
in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)
6 Mathematical Problems in Engineering
Note that by (39) it is not difficult to prove that system(45) (46) has a unique solution 119888(119896)119898119895 (119905) 1 le 119895 le 119896 on interval[0 119879] so let us omit the details
Second Term 1198682 By the Cauchy-Schwartz inequality it gives100381610038161003816100381611986821003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905
By means of the convergences in (44) we can deduce theexistence of constant 119872 gt 0 independent of 119896 and 119898 suchthat 119878(119896)119898 (0) = 10038171003817100381710038171199061119896100381710038171003817100381720 + 119886 (1199061119896 1199061119896)+ 120583 (10038171003817100381710038170100381710038171003817100381720) [119886 (1199060119896 1199060119896) + 10038171003817100381710038171198601199060119896100381710038171003817100381720]
le 121198722(64)
for all119898 119896 isin NTherefore from (63) and (64) we can choose 119879 isin (0 119879lowast]
such that
(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)
Finally it follows from (62) (64) and (65) that119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872))+ 1198632 (119872)int119905
0119878(119896)119898 (119904) 119889119904 (67)
By using Gronwallrsquos Lemma (67) yields119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872)) exp (1199051198632 (119872)) le 1198722 (68)
for all 119905 isin [0 119879] for all119898 and 119896Therefore we have119906(119896)119898 isin 119882 (119872119879) forall119898 119896 (69)
Step 3 (limiting process) From (69) there exists a subse-quence of 119906(119896)119898 still so denoted such that
119906(119896)119898 997888rarr 119906119898 in 119871infin (0 119879 119881 cap 1198672) weaklylowast(119896)119898 997888rarr 1199061015840119898 in 119871infin (0 119879 119881) weaklylowast(119896)119898 997888rarr 11990610158401015840119898 in 1198712 (119876119879) weakly119906119898 isin 119882 (119872119879)
(70)
Passing to limit in (43) we have 119906119898 satisfying (40) (41) in1198712(0 119879) On the other hand it follows from (40)1 and (70)4that 11990610158401015840119898 = minus120583119898(119905)119860119906119898 + 119865119898 isin 119871infin(0 119879 1198712) and hence 119906119898 isin1198821(119872 119879) and the proof of Theorem 8 is complete
We will use the result obtained in Theorem 8 and thecompact imbedding theorems to prove the existence anduniqueness of a weak solution of problem (1)ndash(3) Hence weget the main result in this section
8 Mathematical Problems in Engineering
Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate
1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)
Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])
We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem
Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)
(76)
Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840
Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)
sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879)
(78)
Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
Proof of Lemma 2 Embedding 119881 997893rarr 1198671 is continuous andembedding 1198671 997893rarr 1198620(Ω) is compact so embedding 119881 997893rarr1198620(Ω) is compact In what follows we prove (i)ndash(v)
(i) For all V isin 119881 and 119909 isin [120588 1]|V (119909)| = 100381610038161003816100381610038161003816100381610038161003816int119909
120588V119909 (119910) 119889119910100381610038161003816100381610038161003816100381610038161003816 le int1
120588
1003816100381610038161003816V119909 (119910)1003816100381610038161003816 119889119910le radic1 minus 120588 1003817100381710038171003817V1199091003817100381710038171003817 (13)
(ii) For all V isin 119881and 119909 isin [120588 1]V2 (119909) = 100381610038161003816100381610038161003816100381610038161003816int119909
120588V119909 (119910) 1198891199101003816100381610038161003816100381610038161003816100381610038162 le (119909 minus 120588)int119909
120588V2119909 (119910) 119889119910
le (119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 (14)
Integrating over 119909 from 120588 to 1 we obtainV2 = int1
120588V2 (119909) 119889119909 le int1
120588(119909 minus 120588) 1003817100381710038171003817V11990910038171003817100381710038172 119889119909
= (1 minus 120588)22 1003817100381710038171003817V11990910038171003817100381710038172 (15)
(iii) For all V isin 119881V0 le V le 1 minus 120588radic2 1003817100381710038171003817V1199091003817100381710038171003817 le 1 minus 120588radic2120588 1003817100381710038171003817V11990910038171003817100381710038170 (16)
le 2 V20 + 2 V0 1003817100381710038171003817V11990910038171003817100381710038170 le 2 V20 + V20 + 1003817100381710038171003817V119909100381710038171003817100381720le 3 V21 (18)
implying (v)
Lemma 2 is proved
Remark 3 On 1198712 two norms V 997891rarr V and V 997891rarr V0 areequivalent So are two norms V 997891rarr V1198671 and V 997891rarr V1 on1198671and five norms V 997891rarr V1198671 V 997891rarr V1 V 997891rarr V119909 V 997891rarr V1199090and V 997891rarr radicV11990920 + V2(1) on 119881
where 120577 ge 0 is a constantLemma 4 Symmetric bilinear form 119886(sdot sdot) defined by (19) iscontinuous on 119881 times 119881 and coercive on 119881 that is
(i) |119886(119906 V)| le 11986211199061V1(ii) 119886(V V) ge 1198620V21
for all 119906 V isin 119881 where 1198620 = (12)min1 2120588(1 minus 120588)2 and1198621 = 1 + 3120577Proof of Lemma 4 (i) Byradic1 minus 120588V119909 ge V1198620(Ω) ge |V(1)| andradic120588V119909 le V1199090 for all V isin 119881 we have
ge 1003817100381710038171003817V119909100381710038171003817100381720 = 12 1003817100381710038171003817V119909100381710038171003817100381720 + 12 1003817100381710038171003817V119909100381710038171003817100381720ge 12 1003817100381710038171003817V119909100381710038171003817100381720 + 12 2120588(1 minus 120588)2 V20ge 12 min1 2120588(1 minus 120588)2V21
(21)
Lemma 4 is proved
Lemma 5 There exists Hilbert orthonormal base 119908119895 of space1198712 consisting of eigenfunctions119908119895 corresponding to eigenvalues120582119895 such that
(i) 0 lt 1205821 le 1205822 le sdot sdot sdot le 120582119895 le 120582119895+1 le sdot sdot sdot lim119895rarr+infin120582119895 =+infin(ii) 119886(119908119895 V) = 120582119895⟨119908119895 V⟩ for all V isin 119881 119895 = 1 2 Furthermore sequence 119908119895radic120582119895 is also the Hilbert
orthonormal base of 119881 with respect to scalar product 119886(sdot sdot)
4 Mathematical Problems in Engineering
On the other hand we also have119908119895 satisfying the followingboundary value problem
Proof The proof of Lemma 5 can be found in [17 p 87Theorem 77] with119867 = 1198712 and 119886(sdot sdot) as defined by (19)
We also note that operator119860 119881 rarr 1198811015840 in (22) is uniquelydefined by Lax-Milgramrsquos lemma that is119886 (119906 V) = ⟨119860119906 V⟩ forall119906 V isin 119881 (23)
Lemma 6 On 119881 cap 1198672 three norms V 997891rarr V1198672 V 997891rarr V2 =radicV20 + V11990920 + V11990911990920 and V 997891rarr V2lowast = radicV11990920 + 119860V20 areequivalent
Proof of Lemma 6 (i) It is easy to see that on 119881 cap 1198672 twonorms V 997891rarr V1198672 V 997891rarr V2 = radicV20 + V11990920 + V11990911990920 areequivalent becauseradic120588 V1198672 le V2 le V1198672 forallV isin 1198672 (24)
(ii) For all 119909 isin [120588 1] and V isin 119881 cap 1198672 we have
This implies1003817100381710038171003817119906119909119909100381710038171003817100381720 le 2 11986011990620 + 61205882 1003817100381710038171003817119906119909100381710038171003817100381720le 2(1 + 31205882) (11986011990620 + 1003817100381710038171003817119906119909100381710038171003817100381720)le 2(1 + 31205882) 11990622lowast
(28)
By V0 le ((1 minus 120588)radic2120588)V1199090 for all V isin 119881 we have11990622 = 11990620 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 minus 120588)22120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 + (1 minus 120588)22120588 ) 11990622lowast + 2(1 + 31205882) 11990622lowast= ((1 minus 120588)22120588 + 3 + 61205882)11990622lowast
(29)
(b) Proof 1199062lowast le const1199062It follows from (25) that
Remark 7 The weak formulation of initial-boundary valueproblem (1)ndash(3) can be given in the following manner find119906 isin 119882 = 119906 isin 119871infin(0 119879 119881 cap 1198672) 119906119905 isin 119871infin(0 119879 119881) 119906119905119905 isin119871infin(0 119879 1198712) such that 119906 satisfies the following variationalequation ⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (34)
Mathematical Problems in Engineering 5
for all V isin 119881 ae 119905 isin (0 119879) together with the initialconditions 119906 (0) = 0119906119905 (0) = 1 (35)
where 119886(sdot sdot) is the symmetric bilinear form on 119881 defined by(19)
3 The Existence and Uniqueness Theorem
Now we shall consider problem (1)ndash(3) with constant 120577 ge 0and make the following assumptions
(1198671) 0 isin 119881 cap 1198672 1 isin 119881(1198672) 120583 isin 1198621(R+) with 120583(119911) ge 120583lowast gt 0 forall119911 isin R+
(1198673) 119891 isin 1198620(Ω times R+ times R3) such that 119891(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+ timesR and119863119894119891 isin 1198620(ΩtimesR+ timesR3) 119894 =1 3 4 5Considering 119879lowast gt 0 fixed and letting 119879 isin (0 119879lowast] and119872 gt 0 we put119872 (120583) = sup
Theorem 8 Let assumptions (1198671)ndash(1198673) hold Then thereexist positive constants 119872 119879 such that the problem (40) (41)has solution 119906119898 isin 1198821(119872 119879)Proof of Theorem 8 It consists of three stepsStep 1 (the Faedo-Galerkin approximation (introduced byLions [18])) Consider basis 119908119895 for 119881 as in Lemma 5 Put
120572(119896)119895 119908119895 997888rarr 0 strongly in 119881 cap11986721199061119896 = 119896sum
119895=1
120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)
The system of (43) can be rewritten in form119888(119896)119898119895 (119905) + 120582119895120583119898 (119905) 119888(119896)119898119895 (119905) = 119865119898119895 (119905) 1 le 119895 le 119896119888(119896)119898119895 (0) = 120572(119896)119895 119888(119896)119898119895 (0) = 120573(119896)119895
(45)
in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)
6 Mathematical Problems in Engineering
Note that by (39) it is not difficult to prove that system(45) (46) has a unique solution 119888(119896)119898119895 (119905) 1 le 119895 le 119896 on interval[0 119879] so let us omit the details
Second Term 1198682 By the Cauchy-Schwartz inequality it gives100381610038161003816100381611986821003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905
By means of the convergences in (44) we can deduce theexistence of constant 119872 gt 0 independent of 119896 and 119898 suchthat 119878(119896)119898 (0) = 10038171003817100381710038171199061119896100381710038171003817100381720 + 119886 (1199061119896 1199061119896)+ 120583 (10038171003817100381710038170100381710038171003817100381720) [119886 (1199060119896 1199060119896) + 10038171003817100381710038171198601199060119896100381710038171003817100381720]
le 121198722(64)
for all119898 119896 isin NTherefore from (63) and (64) we can choose 119879 isin (0 119879lowast]
such that
(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)
Finally it follows from (62) (64) and (65) that119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872))+ 1198632 (119872)int119905
0119878(119896)119898 (119904) 119889119904 (67)
By using Gronwallrsquos Lemma (67) yields119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872)) exp (1199051198632 (119872)) le 1198722 (68)
for all 119905 isin [0 119879] for all119898 and 119896Therefore we have119906(119896)119898 isin 119882 (119872119879) forall119898 119896 (69)
Step 3 (limiting process) From (69) there exists a subse-quence of 119906(119896)119898 still so denoted such that
119906(119896)119898 997888rarr 119906119898 in 119871infin (0 119879 119881 cap 1198672) weaklylowast(119896)119898 997888rarr 1199061015840119898 in 119871infin (0 119879 119881) weaklylowast(119896)119898 997888rarr 11990610158401015840119898 in 1198712 (119876119879) weakly119906119898 isin 119882 (119872119879)
(70)
Passing to limit in (43) we have 119906119898 satisfying (40) (41) in1198712(0 119879) On the other hand it follows from (40)1 and (70)4that 11990610158401015840119898 = minus120583119898(119905)119860119906119898 + 119865119898 isin 119871infin(0 119879 1198712) and hence 119906119898 isin1198821(119872 119879) and the proof of Theorem 8 is complete
We will use the result obtained in Theorem 8 and thecompact imbedding theorems to prove the existence anduniqueness of a weak solution of problem (1)ndash(3) Hence weget the main result in this section
8 Mathematical Problems in Engineering
Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate
1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)
Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])
We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem
Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)
(76)
Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840
Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)
sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879)
(78)
Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
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[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
Proof The proof of Lemma 5 can be found in [17 p 87Theorem 77] with119867 = 1198712 and 119886(sdot sdot) as defined by (19)
We also note that operator119860 119881 rarr 1198811015840 in (22) is uniquelydefined by Lax-Milgramrsquos lemma that is119886 (119906 V) = ⟨119860119906 V⟩ forall119906 V isin 119881 (23)
Lemma 6 On 119881 cap 1198672 three norms V 997891rarr V1198672 V 997891rarr V2 =radicV20 + V11990920 + V11990911990920 and V 997891rarr V2lowast = radicV11990920 + 119860V20 areequivalent
Proof of Lemma 6 (i) It is easy to see that on 119881 cap 1198672 twonorms V 997891rarr V1198672 V 997891rarr V2 = radicV20 + V11990920 + V11990911990920 areequivalent becauseradic120588 V1198672 le V2 le V1198672 forallV isin 1198672 (24)
(ii) For all 119909 isin [120588 1] and V isin 119881 cap 1198672 we have
This implies1003817100381710038171003817119906119909119909100381710038171003817100381720 le 2 11986011990620 + 61205882 1003817100381710038171003817119906119909100381710038171003817100381720le 2(1 + 31205882) (11986011990620 + 1003817100381710038171003817119906119909100381710038171003817100381720)le 2(1 + 31205882) 11990622lowast
(28)
By V0 le ((1 minus 120588)radic2120588)V1199090 for all V isin 119881 we have11990622 = 11990620 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 minus 120588)22120588 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909100381710038171003817100381720 + 1003817100381710038171003817119906119909119909100381710038171003817100381720le (1 + (1 minus 120588)22120588 ) 11990622lowast + 2(1 + 31205882) 11990622lowast= ((1 minus 120588)22120588 + 3 + 61205882)11990622lowast
(29)
(b) Proof 1199062lowast le const1199062It follows from (25) that
Remark 7 The weak formulation of initial-boundary valueproblem (1)ndash(3) can be given in the following manner find119906 isin 119882 = 119906 isin 119871infin(0 119879 119881 cap 1198672) 119906119905 isin 119871infin(0 119879 119881) 119906119905119905 isin119871infin(0 119879 1198712) such that 119906 satisfies the following variationalequation ⟨119906119905119905 (119905) V⟩ + 120583 (119906 (119905)20) 119886 (119906 (119905) V)= ⟨119891 (119909 119905 119906 119906119909 119906119905) V⟩ (34)
Mathematical Problems in Engineering 5
for all V isin 119881 ae 119905 isin (0 119879) together with the initialconditions 119906 (0) = 0119906119905 (0) = 1 (35)
where 119886(sdot sdot) is the symmetric bilinear form on 119881 defined by(19)
3 The Existence and Uniqueness Theorem
Now we shall consider problem (1)ndash(3) with constant 120577 ge 0and make the following assumptions
(1198671) 0 isin 119881 cap 1198672 1 isin 119881(1198672) 120583 isin 1198621(R+) with 120583(119911) ge 120583lowast gt 0 forall119911 isin R+
(1198673) 119891 isin 1198620(Ω times R+ times R3) such that 119891(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+ timesR and119863119894119891 isin 1198620(ΩtimesR+ timesR3) 119894 =1 3 4 5Considering 119879lowast gt 0 fixed and letting 119879 isin (0 119879lowast] and119872 gt 0 we put119872 (120583) = sup
Theorem 8 Let assumptions (1198671)ndash(1198673) hold Then thereexist positive constants 119872 119879 such that the problem (40) (41)has solution 119906119898 isin 1198821(119872 119879)Proof of Theorem 8 It consists of three stepsStep 1 (the Faedo-Galerkin approximation (introduced byLions [18])) Consider basis 119908119895 for 119881 as in Lemma 5 Put
120572(119896)119895 119908119895 997888rarr 0 strongly in 119881 cap11986721199061119896 = 119896sum
119895=1
120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)
The system of (43) can be rewritten in form119888(119896)119898119895 (119905) + 120582119895120583119898 (119905) 119888(119896)119898119895 (119905) = 119865119898119895 (119905) 1 le 119895 le 119896119888(119896)119898119895 (0) = 120572(119896)119895 119888(119896)119898119895 (0) = 120573(119896)119895
(45)
in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)
6 Mathematical Problems in Engineering
Note that by (39) it is not difficult to prove that system(45) (46) has a unique solution 119888(119896)119898119895 (119905) 1 le 119895 le 119896 on interval[0 119879] so let us omit the details
Second Term 1198682 By the Cauchy-Schwartz inequality it gives100381610038161003816100381611986821003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905
By means of the convergences in (44) we can deduce theexistence of constant 119872 gt 0 independent of 119896 and 119898 suchthat 119878(119896)119898 (0) = 10038171003817100381710038171199061119896100381710038171003817100381720 + 119886 (1199061119896 1199061119896)+ 120583 (10038171003817100381710038170100381710038171003817100381720) [119886 (1199060119896 1199060119896) + 10038171003817100381710038171198601199060119896100381710038171003817100381720]
le 121198722(64)
for all119898 119896 isin NTherefore from (63) and (64) we can choose 119879 isin (0 119879lowast]
such that
(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)
Finally it follows from (62) (64) and (65) that119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872))+ 1198632 (119872)int119905
0119878(119896)119898 (119904) 119889119904 (67)
By using Gronwallrsquos Lemma (67) yields119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872)) exp (1199051198632 (119872)) le 1198722 (68)
for all 119905 isin [0 119879] for all119898 and 119896Therefore we have119906(119896)119898 isin 119882 (119872119879) forall119898 119896 (69)
Step 3 (limiting process) From (69) there exists a subse-quence of 119906(119896)119898 still so denoted such that
119906(119896)119898 997888rarr 119906119898 in 119871infin (0 119879 119881 cap 1198672) weaklylowast(119896)119898 997888rarr 1199061015840119898 in 119871infin (0 119879 119881) weaklylowast(119896)119898 997888rarr 11990610158401015840119898 in 1198712 (119876119879) weakly119906119898 isin 119882 (119872119879)
(70)
Passing to limit in (43) we have 119906119898 satisfying (40) (41) in1198712(0 119879) On the other hand it follows from (40)1 and (70)4that 11990610158401015840119898 = minus120583119898(119905)119860119906119898 + 119865119898 isin 119871infin(0 119879 1198712) and hence 119906119898 isin1198821(119872 119879) and the proof of Theorem 8 is complete
We will use the result obtained in Theorem 8 and thecompact imbedding theorems to prove the existence anduniqueness of a weak solution of problem (1)ndash(3) Hence weget the main result in this section
8 Mathematical Problems in Engineering
Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate
1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)
Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])
We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem
Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)
(76)
Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840
Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)
sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879)
(78)
Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
Theorem 8 Let assumptions (1198671)ndash(1198673) hold Then thereexist positive constants 119872 119879 such that the problem (40) (41)has solution 119906119898 isin 1198821(119872 119879)Proof of Theorem 8 It consists of three stepsStep 1 (the Faedo-Galerkin approximation (introduced byLions [18])) Consider basis 119908119895 for 119881 as in Lemma 5 Put
120572(119896)119895 119908119895 997888rarr 0 strongly in 119881 cap11986721199061119896 = 119896sum
119895=1
120573(119896)119895 119908119895 997888rarr 1 strongly in 119881 (44)
The system of (43) can be rewritten in form119888(119896)119898119895 (119905) + 120582119895120583119898 (119905) 119888(119896)119898119895 (119905) = 119865119898119895 (119905) 1 le 119895 le 119896119888(119896)119898119895 (0) = 120572(119896)119895 119888(119896)119898119895 (0) = 120573(119896)119895
(45)
in which 119865119898119895 (119905) = ⟨119865119898 (119905) 119908119895⟩ 1 le 119895 le 119896 (46)
6 Mathematical Problems in Engineering
Note that by (39) it is not difficult to prove that system(45) (46) has a unique solution 119888(119896)119898119895 (119905) 1 le 119895 le 119896 on interval[0 119879] so let us omit the details
Second Term 1198682 By the Cauchy-Schwartz inequality it gives100381610038161003816100381611986821003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905
By means of the convergences in (44) we can deduce theexistence of constant 119872 gt 0 independent of 119896 and 119898 suchthat 119878(119896)119898 (0) = 10038171003817100381710038171199061119896100381710038171003817100381720 + 119886 (1199061119896 1199061119896)+ 120583 (10038171003817100381710038170100381710038171003817100381720) [119886 (1199060119896 1199060119896) + 10038171003817100381710038171198601199060119896100381710038171003817100381720]
le 121198722(64)
for all119898 119896 isin NTherefore from (63) and (64) we can choose 119879 isin (0 119879lowast]
such that
(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)
Finally it follows from (62) (64) and (65) that119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872))+ 1198632 (119872)int119905
0119878(119896)119898 (119904) 119889119904 (67)
By using Gronwallrsquos Lemma (67) yields119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872)) exp (1199051198632 (119872)) le 1198722 (68)
for all 119905 isin [0 119879] for all119898 and 119896Therefore we have119906(119896)119898 isin 119882 (119872119879) forall119898 119896 (69)
Step 3 (limiting process) From (69) there exists a subse-quence of 119906(119896)119898 still so denoted such that
119906(119896)119898 997888rarr 119906119898 in 119871infin (0 119879 119881 cap 1198672) weaklylowast(119896)119898 997888rarr 1199061015840119898 in 119871infin (0 119879 119881) weaklylowast(119896)119898 997888rarr 11990610158401015840119898 in 1198712 (119876119879) weakly119906119898 isin 119882 (119872119879)
(70)
Passing to limit in (43) we have 119906119898 satisfying (40) (41) in1198712(0 119879) On the other hand it follows from (40)1 and (70)4that 11990610158401015840119898 = minus120583119898(119905)119860119906119898 + 119865119898 isin 119871infin(0 119879 1198712) and hence 119906119898 isin1198821(119872 119879) and the proof of Theorem 8 is complete
We will use the result obtained in Theorem 8 and thecompact imbedding theorems to prove the existence anduniqueness of a weak solution of problem (1)ndash(3) Hence weget the main result in this section
8 Mathematical Problems in Engineering
Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate
1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)
Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])
We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem
Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)
(76)
Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840
Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)
sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879)
(78)
Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
Note that by (39) it is not difficult to prove that system(45) (46) has a unique solution 119888(119896)119898119895 (119905) 1 le 119895 le 119896 on interval[0 119879] so let us omit the details
Second Term 1198682 By the Cauchy-Schwartz inequality it gives100381610038161003816100381611986821003816100381610038161003816 = 2 10038161003816100381610038161003816100381610038161003816int119905
By means of the convergences in (44) we can deduce theexistence of constant 119872 gt 0 independent of 119896 and 119898 suchthat 119878(119896)119898 (0) = 10038171003817100381710038171199061119896100381710038171003817100381720 + 119886 (1199061119896 1199061119896)+ 120583 (10038171003817100381710038170100381710038171003817100381720) [119886 (1199060119896 1199060119896) + 10038171003817100381710038171198601199060119896100381710038171003817100381720]
le 121198722(64)
for all119898 119896 isin NTherefore from (63) and (64) we can choose 119879 isin (0 119879lowast]
such that
(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)
Finally it follows from (62) (64) and (65) that119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872))+ 1198632 (119872)int119905
0119878(119896)119898 (119904) 119889119904 (67)
By using Gronwallrsquos Lemma (67) yields119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872)) exp (1199051198632 (119872)) le 1198722 (68)
for all 119905 isin [0 119879] for all119898 and 119896Therefore we have119906(119896)119898 isin 119882 (119872119879) forall119898 119896 (69)
Step 3 (limiting process) From (69) there exists a subse-quence of 119906(119896)119898 still so denoted such that
119906(119896)119898 997888rarr 119906119898 in 119871infin (0 119879 119881 cap 1198672) weaklylowast(119896)119898 997888rarr 1199061015840119898 in 119871infin (0 119879 119881) weaklylowast(119896)119898 997888rarr 11990610158401015840119898 in 1198712 (119876119879) weakly119906119898 isin 119882 (119872119879)
(70)
Passing to limit in (43) we have 119906119898 satisfying (40) (41) in1198712(0 119879) On the other hand it follows from (40)1 and (70)4that 11990610158401015840119898 = minus120583119898(119905)119860119906119898 + 119865119898 isin 119871infin(0 119879 1198712) and hence 119906119898 isin1198821(119872 119879) and the proof of Theorem 8 is complete
We will use the result obtained in Theorem 8 and thecompact imbedding theorems to prove the existence anduniqueness of a weak solution of problem (1)ndash(3) Hence weget the main result in this section
8 Mathematical Problems in Engineering
Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate
1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)
Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])
We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem
Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)
(76)
Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840
Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)
sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879)
(78)
Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
By means of the convergences in (44) we can deduce theexistence of constant 119872 gt 0 independent of 119896 and 119898 suchthat 119878(119896)119898 (0) = 10038171003817100381710038171199061119896100381710038171003817100381720 + 119886 (1199061119896 1199061119896)+ 120583 (10038171003817100381710038170100381710038171003817100381720) [119886 (1199060119896 1199060119896) + 10038171003817100381710038171198601199060119896100381710038171003817100381720]
le 121198722(64)
for all119898 119896 isin NTherefore from (63) and (64) we can choose 119879 isin (0 119879lowast]
such that
(121198722 + 1198791198631 (119872)) exp (1198791198632 (119872)) le 1198722 (65)
Finally it follows from (62) (64) and (65) that119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872))+ 1198632 (119872)int119905
0119878(119896)119898 (119904) 119889119904 (67)
By using Gronwallrsquos Lemma (67) yields119878(119896)119898 (119905) le 1198722 exp (minus1198791198632 (119872)) exp (1199051198632 (119872)) le 1198722 (68)
for all 119905 isin [0 119879] for all119898 and 119896Therefore we have119906(119896)119898 isin 119882 (119872119879) forall119898 119896 (69)
Step 3 (limiting process) From (69) there exists a subse-quence of 119906(119896)119898 still so denoted such that
119906(119896)119898 997888rarr 119906119898 in 119871infin (0 119879 119881 cap 1198672) weaklylowast(119896)119898 997888rarr 1199061015840119898 in 119871infin (0 119879 119881) weaklylowast(119896)119898 997888rarr 11990610158401015840119898 in 1198712 (119876119879) weakly119906119898 isin 119882 (119872119879)
(70)
Passing to limit in (43) we have 119906119898 satisfying (40) (41) in1198712(0 119879) On the other hand it follows from (40)1 and (70)4that 11990610158401015840119898 = minus120583119898(119905)119860119906119898 + 119865119898 isin 119871infin(0 119879 1198712) and hence 119906119898 isin1198821(119872 119879) and the proof of Theorem 8 is complete
We will use the result obtained in Theorem 8 and thecompact imbedding theorems to prove the existence anduniqueness of a weak solution of problem (1)ndash(3) Hence weget the main result in this section
8 Mathematical Problems in Engineering
Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate
1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)
Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])
We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem
Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)
(76)
Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840
Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)
sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879)
(78)
Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
Theorem 9 Let (1198671)ndash(1198673) hold Then there exist positiveconstants119872 119879 satisfying (64)ndash(66) such that problem (1)ndash(3)has unique weak solution 119906 isin 1198821(119872 119879) Furthermore thelinear recurrent sequence 119906119898 defined by (40) (41) convergesto solution 119906 strongly in space1198821(119879) = V isin 119871infin(0 119879 119881) V1015840 isin119871infin(0 119879 1198712) with estimate
1003817100381710038171003817119906119898 minus 11990610038171003817100381710038171198821(119879) le 1198721 minus 119896119879119896119898119879 forall119898 isin N (71)
Proof of Theorem 9(a) The Existence First we note that1198821(119879) is a Banach spacewith respect to norm V1198821(119879) = V119871infin(0119879119881) + V1015840119871infin(01198791198712)(see Lions [18])
We shall prove that 119906119898 is a Cauchy sequence in1198821(119879)Let 119908119898 = 119906119898+1 minus 119906119898 Then 119908119898 satisfies the variationalproblem
Second Integral 1198692 By (1198672) it is clear to see that1003816100381610038161003816120583119898+1 (119905) minus 120583119898 (119905)1003816100381610038161003816 = 10038161003816100381610038161003816120583 (1003817100381710038171003817119906119898 (119905)100381710038171003817100381720) minus 120583 (1003817100381710038171003817119906119898minus1 (119905)100381710038171003817100381720)10038161003816100381610038161003816le 119872 (120583) 100381610038161003816100381610038161003817100381710038171003817119906119898 (119905)100381710038171003817100381720 minus 1003817100381710038171003817119906119898minus1 (119905)10038171003817100381710038172010038161003816100381610038161003816le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119908119898minus110038171003817100381710038171198821(119879)
(76)
Hence100381610038161003816100381611986921003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0[120583119898+1 (119904) minus 120583119898 (119904)] ⟨119860119906119898 (119904) 1199081015840
Second Integral 1198693 By (1198673) it yields1003817100381710038171003817119865119898+1 (119905) minus 119865119898 (119905)10038171003817100381710038170 le 119870119872 (119891) (1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170+ 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)
sdot (1 minus 120588radic2120588 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170+ 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1 + 1 minus 120588radic2120588 )sdot (1003817100381710038171003817nabla119908119898minus1 (119905)10038171003817100381710038170 + 100381710038171003817100381710038171199081015840
119898minus1 (119905)100381710038171003817100381710038170) le 119870119872 (119891)(1+ 1 minus 120588radic2120588 ) 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879)
(78)
Hence100381610038161003816100381611986931003816100381610038161003816 le 2 10038161003816100381610038161003816100381610038161003816int119905
0⟨119865119898+1 (119904) minus 119865119898 (119904) 1199081015840
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
Using Gronwallrsquos lemma we deduce from (80) that100381710038171003817100381711990811989810038171003817100381710038171198821(119879) le 119896119879 1003817100381710038171003817119908119898minus1
10038171003817100381710038171198821(119879) forall119898 isin N (81)
which implies that10038171003817100381710038171003817119906119898 minus 119906119898+119901
100381710038171003817100381710038171198821(119879) le 10038171003817100381710038171199060 minus 119906110038171003817100381710038171198821(119879) (1 minus 119896119879)minus1 119896119898119879le 1198721 minus 119896119879119896119898119879 forall119898 119901 isin N (82)
It follows that 119906119898 is a Cauchy sequence in1198821(119879)Thenthere exists 119906 isin 1198821(119879) such that119906119898 997888rarr 119906 strongly in 1198821 (119879) (83)
Note that 119906119898 isin 1198821(119872 119879) and then there existssubsequence 119906119898119895 of 119906119898 such that
119906119898119895 997888rarr 119906 in 119871infin (0 119879 119881 cap 1198672) weaklylowast1199061015840119898119895 997888rarr 1199061015840 in 119871infin (0 119879 119881) weaklylowast11990610158401015840119898119895 997888rarr 11990610158401015840 in 1198712 (119876119879) weakly119906 isin 119882 (119872119879)
(84)
We also note that1003817100381710038171003817119865119898 (119905) minus 119891 (119909 119905 119906 119906119909 119906119905)1003817100381710038171003817119871infin(01198791198712)le 119870119872 (119891)(1 + 1 minus 120588radic2120588 ) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (85)
Hence from (83) and (85) we obtain119865119898 (119905) 997888rarr 119891 (119909 119905 119906 119906119909 119906119905)strongly in 119871infin (0 119879 1198712) (86)
On the other hand we have10038161003816100381610038161003816120583119898 (119905) minus 120583 (119906 (119905)20)10038161003816100381610038161003816 le 2119872119872 (120583) 1003817100381710038171003817119908119898minus1 (119905)10038171003817100381710038170le 2119872119872 (120583) 1003817100381710038171003817119906119898minus1 minus 11990610038171003817100381710038171198821(119879) (87)
Hence it follows from (83) and (87) that120583119898 (119905) 997888rarr 120583 (119906 (119905)20) strongly in 119871infin (0 119879) (88)
Finally passing to limit in (40) (41) as 119898 = 119898119895 rarr infinit implies from (83) (84)13 (86) and (88) that there exists119906 isin 119882(119872119879) satisfying
and thus we have 119906 isin 1198821(119872 119879) The existence of a weaksolution of problem (1)ndash(3) is proved(b) The Uniqueness Let 1199061 1199062 isin 1198821(119872 119879) be two weaksolutions of problem (1)ndash(3) Then 119906 = 1199061 minus 1199062 satisfies thevariational problem
0⟨1198651 (119904) minus 1198652 (119904) 1199061015840 (119904)⟩ 119889119904
minus 2int119905
0[1205831 (119904) minus 1205832 (119904)] ⟨1198601199061 (119904) 1199061015840 (119904)⟩ 119889119904
(93)
with 119885(119905) = 1199061015840(119905)20 + 1205831(119905)119886(119906(119905) 119906(119905))Putting 119870lowast
119872 = 2[119870119872(119891)(1 + (1 minus 120588)radic2120588)(1 + 1radic120583lowast) +(1120583lowast +2(1 minus120588)radic2120583lowast120588)1198722119872(120583)] it follows from (93) that
Using Gronwallrsquos lemma it follows that 119885(119905) equiv 0 that is1199061 equiv 1199062Therefore Theorem 9 is proved
10 Mathematical Problems in Engineering
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
4 Asymptotic Expansion of the Solutionwith respect to a Small Parameter
In this section let (1198671)ndash(1198674) hold We make more thefollowing assumptions
(11986710158402) 1205831 isin 1198621(R+) with 1205831(119911) ge 0 forall119911 isin R+
(11986710158403) 1198911 isin 1198620(Ω times R+ times R3) such that 1198911(120588 119905 0 1199102 0) =0 forall(119905 1199102) isin R+timesR and1198631198941198911 isin 1198620(ΩtimesR+timesR3) 119894 =1 3 4 5Considering the following perturbed problem where 120576 is
First we note that if functions 120583 1205831 119891 1198911 satisfy (1198672)(11986710158402) (1198673) (1198671015840
3) then a priori estimates of the Galerkinapproximation sequence 119906(119896)119898 for problem (1)ndash(3) corre-sponding to 120583 = 120583120576 119891 = 119865120576[119906] |120576| le 1 leads to119906(119896)119898 isin 1198821(119872 119879) where constants 119872 119879 independent of 120576are chosen as in (63)ndash(66) in which 120583 119872(120583) 119870119872(119891) arereplaced with 120583 + 1205831 119872(120583) + 119872(1205831) 119870119872(119891) + 119870119872(1198911)respectively Hence limit 119906120576 in suitable function spaces ofsequence 119906(119896)119898 as 119896 rarr +infin after119898 rarr +infin is a unique weaksolution of problem (119875120576) satisfying 119906120576 isin 1198821(119872 119879)
We can prove in a manner similar to the proof ofTheorem 9 that limit 1199060 in suitable function spaces of family119906120576 as 120576 rarr 0 is a unique weak solution of problem (1198750)(corresponding to 120576 = 0) satisfying 1199060 isin 1198821(119872 119879)
Next we shall study the asymptotic expansion of solution119906120576 with respect to a small parameter 120576 For multi-index 120572 =(1205721 120572119873) isin Z119873+ and 119909 = (1199091 119909119873) isin R119873 we put|120572| = 1205721 + sdot sdot sdot + 120572119873 120572 = 1205721 sdot sdot sdot 120572119873120572 120573 isin Z
119873+ 120572 le 120573 lArrrArr 120572119894 le 120573119894 forall119894 = 1 119873119909120572 = 11990912057211 sdot sdot sdot 119909120572119873119873 (96)
We need the following lemma
Lemma 10 Let 119898119873 isin N and = (1199091 119909119873) isin R119873 120576 isin RThen
[119873 119909] 119898 le 119896 le 119898119873 depending on119909 = (1199091 119909119873) are defined by the following formulas119875(1)119896 [119873 119909] = 119909119896 1 le 119896 le 119873
119875(119898)119896 [119873 119909] = sum
120572isin119860(119898)119896
(119873)
119898120572 119909120572 119898 le 119896 le 119898119873 119898 ge 2119860(119898)
Let 1199060 be a unique weak solution of problem (1198750) that is119906101584010158400 + 120583 [1199060] 1198601199060 = 119891 [1199060] equiv 1198650 120588 lt 119909 lt 1 0 lt 119905 lt 1198791199060 (120588 119905) = 1199060119909 (1 119905) + 1205771199060 (1 119905) = 01199060 (119909 0) = 0 (119909) 11990610158400 (119909 0) = 1 (119909) 1199060 isin 1198821 (119872 119879) (1198750)
Let us consider the sequence of weak solutions 119906119896 1 le119896 le 119873 defined by the following problems11990610158401015840119896 + 120583 [1199060] 119860119906119896 = 119865119896 120588 lt 119909 lt 1 0 lt 119905 lt 119879119906119896 (120588 119905) = 119906119896119909 (1 119905) + 120577119906119896 (1 119905) = 0119906119896 (119909 0) = 1199061015840119896 (119909 0) = 0 119906119896 isin 1198821 (119872 119879) (119896)
where 119865119896 1 le 119896 le 119873 are defined by the following formulas119865119896 = 1198911 [1199060] + Φ1 [119873 119891 1199060 ] minus (1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 1198601199060 119896 = 1 (99a)
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 2 le 119896 le 119873sum119895le119896
⟨119906119895 (119905) 119906119896minus119895 (119905)⟩ 119873 + 1 le 119896 le 2119873(101)
Then we have the following theorem
Theorem 11 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexist constants 119872 gt 0 and 119879 gt 0 such that for every 120576 isin[minus1 1] problem (119875120576) has unique weak solution 119906120576 isin 1198821(119872 119879)satisfying the asymptotic estimation up to order119873+1 as follows
1003817100381710038171003817100381710038171003817100381710038171003817119906120576 minus 119873sum119896=0
11990611989612057611989610038171003817100381710038171003817100381710038171003817100381710038171198821(119879) le 119862119879 |120576|119873+1 (102)
where functions 119906119896 0 le 119896 le 119873 are the weak solutions ofproblems (1198750) (119896) 1 le 119896 le 119873 respectively and 119862119879 is aconstant depending only on 119873 119879 120588 120577 119891 1198911 120583 1205831 119906119896 0 le119896 le 119873
In order to prove Theorem 11 we need the followinglemmas
Lemma 12 Let Φ119896[119873 119891 1199060 ] 1 le 119896 le 119873 be the functionsdefined by the formulas (100) Put ℎ = sum119873
with 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 119891 119906119896 0 le 119896 le 119873
Proof of Lemma 12 In the case of 119873 = 1 the proof of (103)is easy hence we omit the details and we only prove with119873 ge 2 Put ℎ = 1199060 + sum119873
119896=1 119906119896120576119896 equiv 1199060 + ℎ1 By using Taylorrsquosexpansion of function119891[ℎ] = 119891[1199060+ℎ1] = 119891(119909 119905 1199060+ℎ1 11990610158400+
12 Mathematical Problems in Engineering
ℎ10158401 nabla1199060 + nablaℎ1) around point [1199060] equiv (119909 119905 1199060 11990610158400 nabla1199060) up toorder119873 + 1 we obtain
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le119873 in the function space 119871infin(0 1198791198671) we obtain from (100)(105) (110) that 119877119873[119891 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is aconstant depending only on 119873 119879 119891 119906119896 0 le 119896 le 119873 ThusLemma 12 is proved
Lemma 13 Let Φ119896[119873 120583 1199060 ] 1 le 119896 le 119873 be the functionsdefined by formulas (101) Put ℎ = sum119873
119896=0 119906119896120576119896 and then we have120583 [ℎ] = 120583 [1199060] + 119873sum
with 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where 119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873
Proof of Lemma 13 In the case of 119873 = 1 the proof of (111)is easy hence we omit the details and we only prove with119873 ge 2
Mathematical Problems in Engineering 13
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
Put 120585 = ℎ(119905)20 minus 1199060(119905)20 By using Taylorrsquos expansion offunction 120583[ℎ] = 120583(ℎ(119905)20) = 120583(1199060(119905)20 + 120585) around point1199060(119905)20 up to order119873 + 1 we obtain120583 [ℎ] = 120583 (ℎ (119905)20) = 120583 (10038171003817100381710038171199060 (119905)100381710038171003817100381720 + 120585)
On the other hand we also get120585 = 10038171003817100381710038171199060 (119905) + ℎ1 (119905)100381710038171003817100381720 minus 10038171003817100381710038171199060 (119905)100381710038171003817100381720= 2 ⟨1199060 (119905) ℎ1 (119905)⟩ + 1003817100381710038171003817ℎ1 (119905)100381710038171003817100381720 = sum1le119896le2119873
120590119896120576119896 (114)
where 120590119896 1 le 119896 le 2119873 are defined by (101)Using formula (97) again it follows from (114) that
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873in function space 119871infin(0 1198791198671) we obtain from (101) (113)(117) that 119873[120583 1199060 120576]119871infin(01198791198712) le 119862 where119862 is a constantdepending only on119873 119879 120583 119906119896 0 le 119896 le 119873 Thus Lemma 13is proved
Remark 14 Lemmas 12 and 13 are a generalization of theformula contained in [19 p 262 formula (438)] and it isuseful to obtain Lemma 15 below These lemmas are the keyto establish the asymptotic expansion of weak solution 119906120576 oforder119873 + 1 in small parameter 120576
Let 119906 = 119906120576 isin 1198821(119872 119879) be the unique weak solution ofproblem (119875120576)Then V = 119906120576 minus sum119873
119896=0 119906119896120576119896 equiv 119906120576 minus ℎ satisfies theproblem
119864120576 (119909 119905) = 119891 [ℎ] minus 119891 [1199060] + 1205761198911 [ℎ]minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ minus 119873sum
119896=1
119865119896120576119896 (119)
Lemma 15 Let (1198671) (119867(119873)2 ) and (119867(119873)
3 ) hold Then thereexists constant 119862lowast such that1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (120)
where 119862lowast is a constant depending only on 119873 119879 119891 1198911 120583 1205831119906119896 0 le 119896 le 119873
14 Mathematical Problems in Engineering
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
Proof of Lemma 15 We only prove with119873 ge 2By using formula (103) for function 1198911[ℎ] we obtain1198911 [ℎ] = 1198911 [1199060] + 119873minus1sum
where 119877119873minus1[1198911 1199060 120576]119871infin(01198791198712) le 119862 with constant 119862depending only on119873 119879 1198911 1198921 119906119896 0 le 119896 le 119873
By (121) we rewrite 1205761198911[ℎ] as follows1205761198911 [ℎ] = 1205761198911 [1199060] + 119873sum
[1198911 1199060 120576] is bounded in function space 119871infin(0 119879 1198712) by aconstant depending only on119873 119879 119891 1198911 119906119896 0 le 119896 le 119873
On the other hand we put 1205781 = 1205831[1199060] + Φ1[119873 120583 1199060 ]120578119896 = Φ119896[119873 120583 1199060 ] + Φ119896minus1[119873 minus 1 1205831 1199060 ] 2 le 119896 le 119873 andwe deduce from (111) that
minus (120583 [ℎ] minus 120583 [1199060] + 1205761205831 [ℎ]) 119860ℎ = minus119860ℎ[(1205831 [1199060]+ Φ1 [119873 120583 1199060 ]) 120576+ 119873sum
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
By the boundedness of functions 119906119896 1199061015840119896 nabla119906119896 1 le 119896 le 119873 infunction space 119871infin(0 1198791198671) we obtain from (125) (123) and(126) that 1003817100381710038171003817119864120576
1003817100381710038171003817119871infin(01198791198712) le 119862lowast |120576|119873+1 (127)
where 119862lowast is a constant depending only on119873 119879 120588 120577 119891 1198911 1205831205831 119906119896 0 le 119896 le 119873The proof of Lemma 15 is complete
Proof of Theorem 11 Consider sequence V119898 defined by
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
1198722(1205831)](119873 + 1)2(2119873 + 3)21198724((1 minus 120588)2120588)By using Gronwallrsquos lemma we deduce from (140) that1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120590119879 1003817100381710038171003817V119898minus1
10038171003817100381710038171198821(119879) + 120575119879 (120576) forall119898 ge 1 (141)
where 120590119879 = 120578119879radic120588119872 120575119879(120576) = 120578119879119862lowast|120576|119873+1 120578119879 = (1 +1radic120583lowast1198620)radic119879 exp[119879(5 + (1120583lowast)1205902(119872))]We can assume that120590119879 lt 1 with the suitable constant 119879 gt 0 (142)
We require the following lemma and its proof is imme-diate so we omit the details
Lemma 16 Let sequence 120577119898 satisfy120577119898 le 120590120577119898minus1 + 120575 forall119898 ge 1 1205770 = 0 (143)
where 0 le 120590 lt 1 120575 ge 0 are the given constants Then
120577119898 le 120575(1 minus 120590) forall119898 ge 1 (144)
Applying Lemma 16 with 120577119898 = V1198981198821(119879) 120590 = 120590119879 lt 1120575 = 120575119879(120576) = 119862lowast120578119879|120576|119873+1 it follows from (144) that
1003817100381710038171003817V11989810038171003817100381710038171198821(119879) le 120575119879 (120576)1 minus 120590119879 = 119862119879 |120576|119873+1 (145)
where 119862119879 = 120578119879119862lowast(1 minus 120578119879radic120588119872)On the other hand linear recurrent sequence V119898defined
by (128) converges strongly in space 1198821(119879) to solution V ofproblem (118) Hence letting119898 rarr +infin in (145) we get
V1198821(119879) le 119862119879 |120576|119873+1 (146)
This implies (102) The proof of Theorem 11 is complete
Competing Interests
The authors declare that they have no competing interests
Authorsrsquo Contributions
All authors contributed equally in this article They read andapproved the final manuscript
References
[1] G F Carrier ldquoOn the nonlinear vibrations problem of elasticstringrdquo Quarterly of Applied Mathematics vol 3 pp 157ndash1651945
[2] N A Larkin ldquoGlobal regular solutions for the nonhomoge-neous Carrier equationrdquoMathematical Problems in EngineeringTheory Methods and Applications vol 8 no 1 pp 15ndash31 2002
[3] MM Cavalcanti V NDomingos Cavalcanti and J A SorianoldquoGlobal existence and uniform decay rates for the KirchhoffmdashCarrier equation with nonlinear dissipationrdquo Advances in Dif-ferential Equations vol 6 no 6 pp 701ndash730 2001
[4] M M Cavalcanti V N Domingos Cavalcanti and J ASoriano ldquoGlobal existence and asymptotic stability for thenonlinear and generalized damped extensible plate equationrdquoCommunications inContemporaryMathematics vol 6 no 5 pp705ndash731 2004
[5] Y Ebihara L AMedeiros andMMMiranda ldquoLocal solutionsfor a nonlinear degenerate hyperbolic equationrdquo NonlinearAnalysis Theory Methods amp Applications vol 10 no 1 pp 27ndash40 1986
[6] M M Miranda and L P S G Jutuca ldquoExistence and boundarystabilization of solutions for the Kirchhoff equationrdquo Commu-nications in Partial Differential Equations vol 24 no 9-10 pp1759ndash1800 1999
[7] I Lasiecka and J Ong ldquoGlobal solvability and uniform decaysof solutions to quasilinear equation with nonlinear boundarydissipationrdquo Communications in Partial Differential Equationsvol 24 no 11-12 pp 2069ndash2107 1999
[8] M Hosoya and Y Yamada ldquoOn some nonlinear wave equationI local existence and regularity of solutionsrdquo Journal ofthe Faculty of Science the University of Tokyo Section IAMathematics vol 38 pp 225ndash238 1991
[9] L A Medeiros ldquoOn some nonlinear perturbation of Kirchhoff-Carrier operatorrdquoComputational and AppliedMathematics vol13 pp 225ndash233 1994
[10] G PMenzala ldquoOn global classical solutions of a nonlinear waveequationrdquo Applicable Analysis vol 10 no 3 pp 179ndash195 1980
[11] J Y Park J J Bae and I H Jung ldquoUniform decay ofsolution for wave equation of Kirchhoff type with nonlinearboundary damping and memory termrdquo Nonlinear AnalysisTheoryMethodsampApplications vol 50 no 7 pp 871ndash884 2002
[12] J Y Park and J J Bae ldquoOn coupled wave equation of Kirchhofftype with nonlinear boundary damping and memory termrdquoApplied Mathematics and Computation vol 129 no 1 pp 87ndash105 2002
[13] T N Rabello M C C Vieira C L Frota and L A MedeirosldquoSmall vertical vibrations of strings with moving endsrdquo RevistaMatematica Complutense vol 16 no 1 pp 179ndash206 2003
[14] M L Santos J Ferreira D C Pereira andCA Raposo ldquoGlobalexistence and stability for wave equation of Kirchhoff type withmemory condition at the boundaryrdquoNonlinearAnalysisTheoryMethods amp Applications vol 54 no 5 pp 959ndash976 2003
[15] L T P Ngoc N A Triet and N T Long ldquoOn a nonlinear waveequation involving the term minus(120597120597119909)(120583(119909 119905 119906 1199061199092)119906119909) linearapproximation and asymptotic expansion of solution in manysmall parametersrdquo Nonlinear Analysis Real World Applicationsvol 11 no 4 pp 2479ndash2510 2010
18 Mathematical Problems in Engineering
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005
[16] L T P Ngoc and N T Long ldquoLinear approximation andasymptotic expansion of solutions inmany small parameters fora nonlinear Kirchhoff wave equation with mixed nonhomoge-neous conditionsrdquoActa ApplicandaeMathematicae vol 112 no2 pp 137ndash169 2010
[17] R E Showater ldquoHilbert space methods for partial differentialequationsrdquo Electronic Journal of Differential Equations Mono-graph vol 1 1994
[18] J L Lions Quelques Methodes de Resolution des Problems auxLimites Non-Lineares Gau-thier-Villars Paris France 1969
[19] N T Long ldquoOn the nonlinear wave equation 119906119905119905 minus119861(119905 1199062 1199061199092)119906119909119909 = 119891(119909 119905 119906 119906119909 119906119905 1199062 1199061199092) associatedwith the mixed homogeneous conditionsrdquo Journal of Math-ematical Analysis and Applications vol 306 no 1 pp 243ndash2682005