Exponential stability of a linear viscoelastic bar with ...

Annali di Matematica pura ed applicata (IV), Vol. CLXXVIII (2000), pp. 45-66

Exponential Stability of a Linear Viscoelastic Bar with Thermal Memory(*).

CLAUDIO GIORGI - MARIA GRAZIA NASO

A b s t r a c t . - In this paper we study a one-dimensional evolution problem arising in the theory of linear thermoviscoelasticity with hereditary heat conduction. Depending on the istanta- neous conductivity Ko, both Coleman-Gurtin (Ko > O) and Gurtin-Pipkin (Ko = O) heat flow theories are involved. In any case, the exponential stability of the corresponding semigroup is proved for a class of memory functions including weakly singular kernels. In order to achieve the exponential decay of the energy, we assume that mechanical and thermal memory kernels decay exponentially for large time.

1. - I n t r o d u c t i o n .

We consider a linear thermoviscoelastic system occupying a fixed bounded domain ~2 in a natural (unstressed) reference configuration, with constant and uniform abso- lute temperature 0 o. At any point x of the body let consider only small variations of the tempera ture and its gradient from equilibrium reference values (namely, small 0 = e - - 0 0 and g = VO), and small deformations with respect to the reference configuration.

Fur ther , we assume that the Cauchy stress tensor T, the rate h at which heat is ab- sorbed per un i t of volume, and the heat f lux vector q are described by the following linearized constitutive equations of convolution type:

(1.1) T(x , t) = GoVu(x, t ) + G'(s) Vu(x, t - s ) d s +Move(x, t)

o

(*) Entrata in Redazione il 23 luglio 1998. Indirizzo degli AA.: CLAUDIO GIORGI: Dipartimento di Elettronica per l'Automazione, Univer-

sit~ degli Studi di Brescia, Via Branze 38, I - 25123 Brescia, Italia; e-mail: giorgi@b- sing.ing.unibs.it; MARIA GRAZIA NASO: Dipartimento di Matematica, Universith Cattolica del Sa- cro Cuore, Via Trieste 17, I - 25121 Brescia, Italia; e-mail: [email protected]

46 CLAUDI0 GIORGI - MARIA GRAZIA NASO: Exponent ial stability, etc.

(1.2) ] h(x, t) = 0 0 B ' V u t ( x , t) + dOt(x , t) + a(s) Or(X, t - s) ds

Q~ o

(1.3)

oo

q(x, t) = - KoV0(x, t) - f k(s) vO<x, t - s) ds

o

where "t = ~/~t and ' = ~/~s. Here and in the sequel Go and G' (s), s ~> 0, are fourth order tensors, Mo, B, Ko and k(s), s I> 0, are second order tensors, Q0, d, a0 and a ' (s) , s ~> 0, are scalars. In addition, we assume that G', a and k vanishe at infinity.

Constitutive equation (1.3) arises in the heat flow theory of Coleman and Gurtin [2] and generalizes the usual Fourier's law. When K0 = 0, it reduces to the linearized equation proposed by Gurtin and Pipkin [9]. The model described by (1.1)-(1.2) is a special case of that proposed by Lazzari and Vuk in [12]. As to (1.3), however, they considered the case k -- 0, only. By paralleling their procedure, in the Appendix we consider the set of restrictions imposed by the Second Law of Thermodynamic on (1.1)-(1.3).

In the sequel, we restrict our attention to a linear thermoviscoelastic bar with thermal memory, occupying the reference configuration ~2 = [0, 1]. Henceforth, let O(x, t), u(x , t) denote respectively the temperature variation and displacement in the point x e [0, l] at time t. By virtue of thermodynamic restrictions (see Appendix), in the one- dimensional case (1.1)-(1.3) reduce to the following constitutive relations

(1.4)

oo

a(x , t) = Goe(x, t) - f g(s) s(x, t - s) ds - yc0(x, t) o

o o

h(x, t) = yu~(x , t) + CoOt(x, t) + aoO(X, t) + f a ' ( s ) O(x, t - s) ds o

q(x, t) = - K o O ~ ( x , t ) - f k(s) O~(x, t - s) ds o

where a is the axial stress, e =u~ the axial deformation, and moreover, constitutive functions and constants appearing into (1.4) are related to (1.1)-(1.3) by obvious relations. Memory kernels g, a ' and k are assumed to be regular functions decaying to zero as time increases, namely

g,k,a'eCl(O, ~)NLI(0, co).

In particular, we define the viscoelastic equil ibrium modulus as follows

r162

G~ = Go - f g ( s ) ds o

and assume that G~, Co and ~, are positive constants.

CLAUDIO GIORGI - MARIA GRAZIA NASO: Exponential stability, etc. 47

(1.5)

For later convenience, equations (1.4) can be written as

co

a(t) = G| e(t) + f g(s)[e(t) - e(t - s)] ds - ~,cO(t) o

h ( t ) = ~ u ~ t ( t ) + c o O t ( t ) + a o O ( t ) + f a ' ( s ) ~s O ( t - v ) dv ds o o

f 8 O ~ ( t - ~) dv ds q ( t l = - K o O ~ ( t ) - k(s) -~So

where the dependence on x is understood and not written. Letting

(1.6)

(1.7)

ut(s) = u(t - s), or(s) = O(t - s)

wt(s) = u(t) - u t ( s ) ,

formally integrating by parts k( ~ ) = 0, it follows

(1.8)

s t

~]t(s) = f o t ( ~ ) dv = f ~ v ) d r , 0 t - s

both integrals in (1.5)2-(1.5)3, and using y t ( 0 ) = 0 ,

0o

a(t) = G~ u~(t) + f g(s) wt(s ) ds - ~,cO(t) o

oo

h(t) = yu~t(t) + CoOt(t) + aoO(t) + ~ v(s) ~lt(s) ds o

o~

q(t) = - KoO~(t) - I tt(s) rit~(s) ds o

where/~ = - k ' and v = - a " . According to (5.10)-(5.11) we have ao, K0 ~> 0. Now, we match (1.8) with motion and energy balance equations involving zero

sources, namely

Q o U t t - - a x ~- 0

~oh + q~ = 0 .

As a consequence, assuming Q 0 = Co = 1, we obtain the following system of integro-par-

48 CLAUDIO GIORGI - MARIA GRAZIA NASO: Exponent ia l stability, etc.

tial differential equations:

utt(t) = G~ u ~ ( t ) + [ g(s) w t (s) ds - ycO~(t) o

(1.9) v~t(t) = Kov~=~(t) + ~tt(s) ~ t ( s ) ds - ~v(s ) ~?t(s) ds - aoO(t) - r u ~ ( t ) o o

wtt(s) = ut( t ) - wt (s ) , s >I 0

~ ( s ) = ~(t) - ~ t (s), s >I O.

Unlike the thermoelastic problem, this system involves the additional variables ~]t and w t, whose evolution equations have been easily obtained from (1.7) by differentia- tion with respect to t. In addition, the above choice of variables requires to specify the initial data at t = 0 on w t and r] t, too. Therefore, initial conditions imposed to (1.9) are as follows

U ( 0 ) --~ U0(X) , u t ( O ) : U I ( x ) , ~ ( 0 ) ---- ~O(X)

u ( - s ) = w~ s >I 0

In connection with boundary conditions at x = 0, l, we assume that both ends are fixed and no heat flows through them (adiabatic thermal conditions), namely

(1.11) u(t) = O, q(t) = 0 at x = O, l, t >I O.

However, when fixed ends under isothermal (v~ = 0) or mixed (0 = O/q = O) thermal boundary conditions are considered, all results in this paper still hold with slight modifications.

Concerning the constitutive m e m o r y keraels g, ~ and v in (1.10), we assume the following set of hypotheses.

(hl) g , ~ , v E C I ( R + ) A L I ( R + ) , g , ~ , v>~O

go-- fg(s) >o, too= f (s)ds>0, no = fv(s)ds>0 o o o

(h3) g ' ( s ) <.0, i t ' ( s ) ~ 0 , u ' ( s ) <~0, for s e R +.

In addition, if we expect to achieve exponential decay of the energy, we must assume that both g,/~ and u decay exponentially as s--~ :r namely there exist three positive constants 51, 5 2, 5 a such that

(h4) - g ' ( s ) >1 (~ l g ( 8 ) , - -~ ' (S ) >I 5~/t(S), for s e R +

(h5) - v ' (s)/> 5 3 v(s), for s e R + .


Problems like (1.9) are not new. Indeed, many efforts are devoted to studying the asymptotic behavior in thermoelasticity and thermoviscoelasticity. In his pioneer work [3], Dafermos considered the thermoelastic problem involving no memory effect ( g - -=/~ - v ---- 0) and proved that the energy E(t) converges to zero asymptotically if the initial state (u0, Vo, v~0) is in Hol (~) x L2(~) • L 2 ( ~ ) and u, v~ satisfy Dirichlet-Dirichlet boundary conditions. Since then, much progress has been made to obtain the exponential decay rate of solutions in thermoelasticity. In this direction, we refer to Slemrod [22], Hansen [10], Mufioz Rivera [15], [17], [19] Kim [11], Liu and Zheng [13], Burns, Liu and Zheng [1].

Recently, the asymptotic behaviour of systems in linear viscoelasticity and thermoviscoelasticity was investigated by many authors. In particular, we mention the works of Mufioz Rivera [16], [18], [20] in which a careful exploitation of some energy-like inequalities is performed. Liu and Zheng [14] proved the exponential stability of the semigroup associated with the integro-partial differential equations (1.9) with/~ - v - 0 (no heat flux and energy memory term). As a particular case, they proved the exponential stability of the linear viscoelastic system (y = 0, h -- q - 0). Fabrizio and Lazzari [4] obtained a similar result in three dimensions using a completely different method based upon the Datko lemma.

When K0 = 0 and one smooth thermal memory kernel/~ is involved, only, it is worth noting that system (1.9) describes the motion and heat propagation in a thermoviscoelastic bar with thermal damping as weak as possible. In addition, adiabatic boundary condition (1.11)2 do not allow the heat produced inside the bar to flow out. The exponential stability of such a system, apparently, has not been treated at all by the present approach. This property is important for the study of the global existence of solutions to the corresponding nonlinear system with small initial data and for the existence of optimal control.

In the sequel we shall separately study the evolution problems corresponding to K0 = 0 and Ko > 0. Section 3 is devoted to the former, section 4 to the latter. In both cas- es our aim is to establish the exponential decay in time of solutions for the problem (1.9)-(1.10). To this end we parallel the procedure followed by Liu and Zheng in [14] and re-cast the heat flux equation with memory in the semigroup framework. However, when Ko = 0 a more carefull analysis is required because the thermal damping is weaker. In this connection we stress that the presence of the thermal memory kernel v is crucial only if adiabatic boundary conditions are involved.

Along this line we also mention the paper [8]. There, the exponential energy decay is established for a rigid, linear, hereditary heat conductor ( y=Ko=ao=O, v - - 0 ) .

2. - F u n c t i o n a l s e t t i n g and n o t a t i o n .

Let s = [ 0, l] r R. With usual notation, we introduce the space L 2, Ho 1 and H - 1 act- ing on ~. Hereafter, (., .> denotes the L 2 inner product, and I1"11 denotes the L 2 norm. Recalling Poincar6 inequality

(2.1) II II 2 llv ll 2 w Ho 1,

50 CLAUDIO GIORGI - MARIA GRAZIA NASO: Exponential stability, etc.

for some 2 o > 0, Ho 1 can be equipped with the inner product

(~, v).~ = (Vu, Vv).

The boundedness of ~ and the assumption

supp v ~ supp/~

enable us to define the following Hilbert spaces

L2(R+; g; H I) = R +--->Ho ~ [g(s)llcfx(s)ll2ds < + o

{ : } L 2 ( R + ; l u ; v ; g l ) = ~ : R + - - > H ' /~(s)llq)~(s)[[2ds+ v(s)ll~(s)lpds< + ~ o o

respectively endowed with the inner products cr

= [g(s)(q~(s), ~ ( s ) ) ds, (r o

oo oo

o o

Finally, for sake of simplicity we let

W = L2(R + ; g; Ho~),

N = {~eL2(R+; ~

and introduce the Hilbert space

; v ; H 1 ) ] f / z ( s ) c f x ( s ) d s = O on a~2 ,

o

X : = U x V x O x W x N

whose inner product is given by

(z, ~)x = (u, ~)u + (v, ~)v + (~, ~)e + (w, ~)w + (7, ~)~ =

= G~(ux, ~ ) + (v, ~) + c(v ~, -~) + [g(s)(w~, ~ ) ds + o

o o

We conclude this introductory part with some basic facts about semigroup of operators. For a detailed exposition of the subject the reader is referred to [21]. In the sequel of this section, let H denote a real Hflbert space endowed with the scalar product (., .) and

CLAUDI0 GIORGI - MARIA GRAZIA NAS0: Exponent ial stability, etc. 51

norm il-II. Since no confusion should occur, we denote again by I1"11 the norm of a bounded operator on H.

THEOREM 2.1 (Lumer-Phillips). - Let ~ be a l inear operator wi th dense domain D(2) in H. I f ~ is dissipative and there is a ~ o > 0 such that the range, R(2 o I - ~), of

o I - ~ is H, then ~ is the inf ini tesimal generator of a Co-semigroup of contractions on H (i.e., a Co-semigroup T(t) such that IIT(t)ll <~ i for every t >I 0).

Finally, for later convenience we recall that a linear Co-semigroup T(t) of contractions is said to be exponentially stable if there exist two constants M I> 1 and fl > 0 such that

(2.2) liT(t) Zo I1~ ~ Me -st I[zollH, VZo e H, Vt > O.

We recall that the complexification of H is the complex Hilbert space Hc, defined by

Hc = { z ] z = x + i y , x, y e H } ,

endowed with the inner product

(xl + iyl , x2 + iY2)c = (xl, x2) + (Yl, Y2) + i(yl , x2) - i(xl , Y2).

Analogously, the complexification ~ of 2 is the linear operator on Hc with domain

D(2_~)= { z l z = x + iy , x, y e D ( 2 ) }

defined by

2c(X + iy) = 2 x + i 2 y

and the corresponding semigroup S(t) on ~ is defined by

S( t ) (x + iy) = T( t )x + iT(t) y .

In order to prove the exponential stability of the semigroup T(t) generated by ~ we shall use the following statement(see [8]).

LEMMA 2.1. - Let T(t) be a contraction semigroup on a real Hilbert space H, let be its inf ini tesimal generator. I f the operator iflI - 2 c is uni formly bounded below as f l e R , that is, i f there exists a > 0 such that

(2.3)

then T(t) is exponentially stable.

52 CLAUDIO GIORGI - MARIA GRAZIA NASO: Exponent ia l stability, etc.

3. - M a i n r e s u l t s .

(3.1)

If we consider the weaker case K0 = 0 in (1.9), we have the following problem

o o

U t t ( t ) = G~ u=,(t) + f g(s) w2As) ds - rcO At) o

Or(t) = - a o v ~ ( t ) - T u f t ( t ) + f ~(s) ~(s ) d s - f v(s) ~'(s) ds o o

w ~ ( s ) = u t ( t ) - w ~ ( s )

~ ( s ) = o ( t ) - ~ ( s ) .

In the adiabatic case, it is worth noting that no boundary condition on 0 turns out, in fact (1.11) simply implies

(3.2) u(t) = O,

o o

f t t ( s ) y t ( s ) d s = O at x = 0 , 1 for t � 9 +. o

The energy of sistem (3.1) is easily seen to be

E(t ) = G~llu~(t)ll 2 + Ilut(t)lr + cllO(t)ll 2 + f g(s)Hwt(s)ll2ds + o

o o ) § f~ ( s ) l l~ ( s ) l l ~ ds + c f v(s)ll~t(s)ll2ds .

o o

After introducing the new variable v(t) = ut(t) , and setting

z = (u, v, O, w, ~])r,

system (3.1) can be written as a linear evolution equation

(3.3) zt(t) = ~ z ( t )

on the Hilbert space X, and its energy is represented by

1 IIz(t)IlZx �9 E( t ) =

CLAUDIO GIORGI - MARIA GRAZIA NASO: Exponent ia l stability, etc. 53

The operator O: in (3.3) is given by

(3.4)

I'u

Iw 1,7

V

G ~ u ~ + f gw~ ds - ~cO o

- ~ , v ~ - ~ . o + I ~,,7x~ds - I v,Tds o o

V - - U s

and its domain is defined as

D(a) = z e X

G ~ u + fg ( s ) w(s) d s e H 2 n H ~ , o

v e i l 1, O e H 1

f /~(s) ~x~(s) ds - f v(s) ~?(s) ds e L ~, o o

W(S) e H I ( R + ; g; Hol), w(0) = 0,

~(s) e H ~ ( R § v; H~), ~(0) = 0

According to (1.10) and letting vo = ul, initial conditions reduce to

(3.5) z(0) = Zo, Zo ~ X

where z0 = (Uo, Vo, 0o, w ~ ~]o).

3.1. Well-posedness.

In the sequel we shall prove the well-posedness of the initial boundary value problem (3.1), (1.11), (3.5). In view of Theorem 2.1 we prove the following Lemmas.

LEMMA 3.1. - I f the memory kernels satisfy (hl)-(h3), then r is dissipative.

PROOF. - For every z e D(O:) we have

(az, z)x = (v, u)u + (G~ u~, + ~ gwx, ds ~c# ~, v +

o V

o o o

+ (v - ws, w)w +

+ (o - , s , , ) ~ = - Ca o ll~ll 2 - (ws, w ) w - ( , s , ~)~.


Integrating by parts it follows that

(3.6)

1 fg,(s)[[wAs)[[Ud s ( w . , = -

o o o o~

( ~ " ~)N= - 2 ~'(s)II'As)II2ds- 2 o o

The above calculation is obtained formally taking product in W and N and can be made rigorous with the use of mollifiers (see [7]). Thus, in view of (h3), we obtain

(a z , Z>x-- - caol/t~ll 2 + ~ g'(s)]lwAs)ll2ds + - ~'(s)]l~lAs)ll2ds + 2

o o

ar

c f )lJ )JJ + - v ' (s rl(s gds<.O 2

o

so proving the dissipativeness of (~. �9

LEMMA 3.2. - Under the same assumptions of Lemma 3.1, the operator I - (2 is surjective.

PROOF. - In order to determine the range of I - (2, we consider the system

( I - r z = ~,

where ~ = (~, ~, ~, ~, ~) e X, namely

(3.7)

U - - W .-~ ~t

o o

v - G~ u~ - fg(s) w~(s) ds + ~cOx = o

o o o ~

+ rv~ + aoO - f i t(s) y~ ( s ) ds + fv (s ) y(s) ds = o o

W - - V + W s = W

Integrating (3.7)4 and (3.7)5 we obtain

(3.8)

B

w(', s) = v(.)(1 - e -8) + f e~-S~(., v) dv o

8

r/(-, s) = v~(.)(1 - e -8) + ~e~-S~(., v) dr . o


Substituting v and w from (8.7)1 and ( 8 . 8 ) 1 into (8.7)2 we obtain

(8.9) u - c g u ~ + y c t ~ = v + u + J g ( s e - 8 - 1 ) g ~ + o e ~ - 8 ~ d v ds,

and substituting ~ and v from (3.8)2 and (3.7)1 into (3.7)3 yields

~ 8 ~ 8

(3.10) cvt~-c~,t~ + 7u~= 0 +Tu~ + f ~ ( s ) f e ~ - 8 ~ d ~ d s - f~(s)fe~-8~ d ~ , 0 0 0 0

where

Cg = G| + fg (s ) (1 - e -8) ds o

c, = f/x(s)(1 - e -s) ds o

= 1 + ao + f v ( s ) ( 1 - e -8) ds. Vv

0

All these constants are positive by virtue of (hl)-(h2). Moreover, it can be shown that the right-hand sides of (8.9)-(8.10) are in H -1

Firs t we prove that u �9 Ho ~ and t~ �9 H 1. After multiplying (8.10) by c, we consider the bilinear form a associated with (8.9)-(8.10). For all y = (Yl, Y2) �9 1 • we obtain

a(y , y) = cgllYl~ II 2 + ec~ Ily~}{ 2 + {{y1112 + verily211 ~ - rc f (y~yl~ + yl y~) d~.

Since the last integral vanishes, there exists a positive constant o such that

a(y, y) z cr(Hyl{{~ d + IlY211~1)

and Lax-Milgram theorem provides existence and uniqueness of the solution

(u, v ~) � 9 1 • H 1

of the problem (8.9)-(3.10). As a consequence, w �9 W and ~/�9 N. In fact, (8.7)1 yields v �9 �9 Ho ~ so that from (8.7)4 and (8.6)1 it easily follows

r162

if Ilw}l~= <w, v)w+ -~ g'(s)llwAs)ll~ds + <w, ~h~ o


which in turn, by virtue of (h2) and Young inequality, yields

1 2 : llwllw goll =ll § II II . 2

In like manner, from (3.7)5 and (3.6)2 we obtain

1 IMI~ <~ cmo I1~ x II 2 + cno I1~112 + II ~ 112N �9 2

From (3.7)2 and (3.7)3, we get that

r

G~ ux~ - [ g(s) w~(s ) ds e L 2 o

and

f lt(s) y ~ d s - f v(s) yds e L 2. o o

Furthermore, from (3.7)4 and (3.'/)5 we have that

Ilw, llw <. gollV~l[ + Ilwllw + Ilwllw

and

Hence ws e W and ~/~ e N. Thus I - ~ is surjective.

IlY sllN ~< C max (too, nO)IIV~IIH1 + IMIN + II~IIN"

As a consequence of Theorem 2.1, ~ is the infinitesimal generator of a Co-semigroup, T(t), of contractions on the Hilbert space X.

3.2. Exponential stability.

In the sequel we take into consideration the asymptotic behavior of solutions z(t) = = T(t)zo of (3.3). In particular, we prove the exponential stability of the semigroup generated by (~ by means of Lemma 2.1.

THEOREM 3.1. - Suppose that kernels g, it and r satisfy conditions f rom (hl) to (h5). Then T(t) = eta is exponentially stable.

PROOF. - Let Xc and (2c be the complexification of X and ~, respectively. Firt, we prove that S(t), the contraction semigroup on Xc generated by Ctc, is exponentially stable. We use the contradiction argument assuming that the conclusion of Lemma 2.1 is not true. Thus, we consider the case when (2.3) fails to hold. Namely, there exists a sequence of fin e R and a sequence of h~ e D ( ~ c ) r such that

(3.11) lim I I ( i f l , I - C ~ ) h n l ~ c = O , IlhnlL~c=l Vn. n - - - > + ~

CLAUDIO GIORGI - MARIA GRAZIA NASO: Exponential stability, e t c . 57

As n--* r162 the limit in (3.11) is equivalent to

(3.12) i ~ n h ( n 1) - h~(2)---)0 in Uc

oo

(3.13) i ~ n h ( n 2) - G~D~h(n 1) - IgD}h(~4)ds + ~,cD~h(n3)-->O in Vc o

oo oo

(3.14) ifl,~h(~ a) + y D ~ h (2) + a o h , (3)- II~D~h(,5)ds+ fvh(~5)ds--->O in Oc o o

i f l n h (a) - h~ (2) + D . h (a) ---> 0 in Wc

ip ,h2 ) - h2 + D.h(J)--,O in

(3.15)

(3.16)

Denoting by II'llc the norm of L~, the complexification of L 2, it follows that

(3.17) R e ( ( i f l n I - Ctc) hn, h~)xc = - Re(~ch~, h~)c =

ao oo

=caollh(na)H5 - -~ g'(s)llD~h(~ 115ds- c r,(s)llh(5)(s)HSd s + o o

oo

c [ )H )]]5 2 /~'(s D~h 5)(s ds---)O.

o

Here, by virtue of (h3), each term is nonnegative and then tends to zero. Moreover, we obtain

(4) _.~ (3.18) Ilh(3)llOc--)o Ilh~ IIwc 0 Ilh~(5)llNc--)0.

Indeed, because of (h4)-(h5), the following inequalities hold

I ' d Ih(4) 2 - g (s)]]D~h(,a)(s)ll~d s>~ 1 ~ we, o

oo oo

- e I v ' (s)llh (nS)(s)ll~ds - c I it ' (s)llD~h (~5)(s)ll2c & >I 5llh(5) II}c o o

where 5 = min{52, d3}. As a consequence of (3.18), we have

(1) 2 (3.19) Ilhn IIUc + IIh(2)ll~c--~ 1.

On the other hand, from (3.12) and Poincar~ inequality, we infer

ifl~h(~ 1) - h~(2)--)0 in Vc = L~,

58 CLAUDIO GIORGI - MARIA GRAZIA NASO: Exponential stability, e t c .

so that

(3.20) ifl n(h~ (1), h(2))c - IIh (~ 2) II~c ~ O .

Starting, from (3.13) we obtain

i~n(h(n 2), h(1))c- G~ (D: h(~ 1), h(1))c +

2 (4) yc(Dxh(n3), h(nl))c - fg(s)(D~ hn (s), h(nl))cds + = 0

oo

(1) 2 ~ 4)(S), + = i~n(h(n 2), h(i))c + Ilhn IlUc + g(s)(Dxh(~ D~h(,1))cds 0

-rc(h (n 8), n h2))c-,O

where the last two terms converge to zero because of (3.18), (3.19) and the following inequality

0

<<- ~ g(s)HDxh (~a)(s)Hc IlDxh (~ 1) Ilcds <- 0

oo

1 iih(1)llucllh(4)llWc ~g(s)ds__>O. G|

0

Thereby

(1) 2 ..._) ifln(h(2), hn(1))c + Ilhn IIUc 0

and adding to it the complex conjugate of (3.20) we get

(1) 2 (3.21) Ilhn Ilyc -IIh.(2)ll~c--)O.

By comparing this limit with (3.19) it follows that

1 1 (3.22) ~(1) 2 __~ ~(2) 2 __>-. ,on Iuc ~ , ,~n Wc 2

Now, we are forced to claim [fin I ~> ~ > 0, for all n. Otherwise, from (3.12) we infer that h~(2)-*0 in Uc (at least a subsequence), so does in Vc. A contradiction. Dividing (3.12) by ifl n and applying (3.22) yields

(3.23) ha(2) ~yc 1

We complete the proof showing that (3.23) leads to a contradiction. Indeed, we rewrite


(3.15) in the form

(3.24) h(4) h~ (2) D8 hn (4) - + *0 i n W c .

Under conditions (hl), (h3) imposed on g(s), by virtue of [14], p. 29, we can easily check that (sh(,2)/iS,)eWc. Thus, the limit (3.24) yields

(3.25) / ) sh(2) _ h(2) 2 f s g ( s ) ds +

1 f h(~ 2) + iS--~ sg(s) 8h(4)(s)' -~n ds-->O. 0 uc

I t is clear from previous argument that the first term in (3.25) converges to zero. To estimate the third term, first we integrate it by parts

- - ds = - sg ' ( s ) h(~4)(s), ds + o iS~ luc iS. luo

- lg(s) h(,4)(s), --h(~2) ds, o , iS . I~

then we estimate each term on the right-hand side to get

( ( | \1/2H

h(2) h,(z) ~g(s) h2)(s), 7 ~ ~ ~ d~-<Slllh~(')ll~

0

where flo = -G~ s2g'(s) ds and fll = (goG~) 1/2. Thus, by virtue of (3.17), (3.18)

and (3.23) the third term of (3.25) converges to zero as well as the first. Thereby, the second term of (3.25) must converge to zero, too, but this contradicts (3.23). As a consequence, S(t) is exponentially stable: there exists a constant oJ > 0 such that

lIs(t) ooiEo-< e-~'ilool~c, vo0 = (x0 + iy0) ~ x c .

60 CLAUDIO GIORGI - MARIA GRAZIA NAS0: Exponen t ia l stabili ty, etc.

In particular, if a o = Zo + i 0 then

IlS(t) ao IL~c = liT(t) ZotIx <<- e - ~ t H a o l l x c = e -'otIIzolL~,

so proving the exponential stability of T. �9

4 . - F u r t h e r r e s u l t s .

In this section we consider the problem (1.9)-(1.11) assuming Ko > 0, namely

(4.1)

oo

uu(t ) = G~ u s ( t ) + f g(s) w t (s) ds - ~ct~ ~(t) o

Or(t) = - aov~(t) - ru~t(t) + K o v ~ ( t ) + f i t ( s ) t l t , ( s ) ds - f v(s) tl t(s) ds o o

w~(s) --- ut(t) w~(s)

rl~(s) O(t) t = - ~ ( s )

with initial conditions (3.5) and boundary conditions

(4.2) u(t) --- O, oo

~(s) K o a , ( t ) + ~(s) ds = o o

at x = 0 , l for t e R +.

In this case, the well posedness of problem (4.1), (4.2), (3.5) follows by paralleling the previous procedure. Indeed, system (4.1) can be written as

(4.3) z t ( t ) = ~ z ( t ) ,

where ;% is defined by

u

v

~X t~

W

V

oo

G| u ~ + f g w ~ d s - ~,ct~ ~ o

o~ o~

o o

v - - w s


and

D(~) = . ~X

G~u + fg(s ) w(s) d s e H 2 NH~, v e i l 1, O e H 1 o

Kov~, + f ~ ( s ) ds - f v(s) ri(s) ds e L 2 o o

w(s) ~HI(R+; g; H1), w(0) = 0

~(s) e H I ( R +;/~; v; H1), tl(0) = 0

First, under assumption (hl) and (h3), we have

if -cKoll ll - c oll ll + g'(s)llw (s)ll d + o

+ - - ,U' (8 TJx(S)H2ds-'l - - "Y'(S ~(s)ll2ds~O 2 2

o o

so proving the dissipativeness of ~ . Moreover, the operator I - ~ is surjective. As in section 3, indeed, we obtain the system (3.9)-(3.10), where

% = G| + fg(s)(1 - e -8) ds > 0 o

c~, = Ko + I/~(s)(1 - e -s) ds > 0 o

cv = 1 + a 0 + fv(s)(1 - e-S) ds > O, o

and Lax-Milgram theorem provides the required result. As a consequence, ~ is the infinitesimal generator of a C0-semigroup T(t) of contractions on the Hilbert space X.

Finally, the exponential stability of the semigroup T(t) generated by ~ is proved by means of Lemma 2.1. Assuming (hl)-(h5) we parallel the proof of Theorem 3.1. Firts, we obtain relations (3.12), (3.13), (3.15), (3.16) and

(4.4) iflnh (a) + ~,D~h (2) + aoh(3) + KoD~h (3) ~ 2 (5) - p ( s ) D ~ h n (s) d s + o

+ I t ( s ) hn(5)(s) ds-->O in Oc o


instead of (3.14). Then, as n--* oo, we observe that

Re((ifl,~I - 9(c) hn, hn)rc = - Re (9(chn, hn)rc =

o o

1 I =cao[[h(n3)[[~ 4- cKo[[D~:h(3)[[2 c - ~ g'(s)]]D=h2)(s)H~ds +

o

co co

c f )ll )11 c f , )Ilk(:> - - - v ( s ( s d s ~ O . 2 # ' ( s Dxh(5)(s d s - 2 o o

Since each term is nonnegative and tends to zero, (h4) and (h5) yield

, h (4) ._-> (4.5) IIh2)llo --)o, IlD h2>ll -->0 II. Ilw o, IIh2>ll o

so that

(4.6) [[h~(1) ][~] c + [[h,(2) []2 c--> 1.

From here on, the proof proceeds as in the weaker case, Ko = 0, with slight modifications. Then, the semigroup generated by Y~ decays exponentially.

REMARK 4.1. - When Ko> 0, it is worth noting that previous results, in particular the exponential decay, still hold when a 0 vanishes and the memory kernel v is removed from the constitutive equation (1.4)2, provided that all boundary conditions are of Dirichlet type. Of course, functional spaces and operator domains have to be rear- ranged accordingly.

REMARK 4.2. - Assuming a temperature boundary condition is of Dirichlet type, the decay condition (h5) on the kernel v is not crucial to prove the exponential stability results for problems (3.1) and (4.1). In fact, by virtue of the Poincar~ inequality (2.1), we can substitute (h3)3 and (h5) by the weaker conditions

~'(s) + t o y ' ( s ) <.0 for s e R +

v'(s)+53v(s)+1o/~'(s)<<.O for s e R +.

5. - A p p e n d i x .

We assume that constitutive equations satisfy the first and second law of thermody- namics, namely (1.1)-(1.3) are such that the power balance

CLAUDIO GIORGI - MARIA GRAZlA NASO: Exponent ial stability, etc. 63

and the Clausius inequality

(5.1) Qoh(t) q( t ) .g( t )

+ Oo + O(t) [0o + ~ t ) ] 2

dt <<. O

hold for any cyclic process. Because of the assumed smallness of v ~, under linear ap- proximation the quantity (00 + v~) -1 in (5.1) may be substituted by its linear Taylor polynomial, thus yielding the approximate Clausiu8 inequality

0~ ~ [ Q o h ( t ) ~ t ) - Q o h ( t ) O o - q ( t ) . g ( t ) ] d t>10.

Now, taking into account the power balance we obtain

(5.2) 0-~o [•oh(t) ~ t ) + OoT(t ) .Vut( t ) - q(t).g(t)] dt >>. O .

After integrating by parts the integral in (1.2) and substituting constitutive equations (1.1)-(1.3) into (5.2), we have the following inequality:

(5.3) ~ GoVu( t )+ G ' ( s ) V u ( t - s ) d s + ( M o + B ) O ( t ) . r u t ( t ) +

o

1 + a o O ( t ) + ~ a ' ( s ) v ~ ( t - ds O( t )+ + dyer(t)

o

+ - - Kog(t) + k(s )g( t - 8) ds .g(t) dt >I O. Oo o

In order to scrutinize the consequences of this statement, we consider time-oscillating histories of strain and temperature. For any given positive constant frequency co, let

ru t ( s ) = Vul cos co(t - s) - Vu2 sin co(t - 8) = Re [rue ~~ 8)] (5.4)

~t(s) = 01 cos co(t - s) - v~2 sin co(t - s) = Re[v~e ~r 8)]

where u = Ul + iu2 and v~ = 01 + ivY2 are constant in time. A cyclic irreversible process ( ru t , v~t, g) with period 2z/co is then defined by:

ru t ( t ) = - co(Vul sin cot + Vu2 cos cot)

(5.5) ~t( t ) = - co(O 1 sin cot + 02 cos cot)

g(t) = gl sin cot + g2 cos cot.

For later convenience we denote by 97 the formal Fourier t ransform of any scalar- or

64 CLAUDI0 GIORGI - MARIA GRAZIA NASO: Exponential stability, etc.

vector-valued causal function f defined on R § with values in banach space, namely

o v

f(w) = ~f(s) e-~Sds w e R , �9

o

Moreover, for any given function g on R +, we formally have

0c(W) : fg(s) coswsds, 08(0) : ~g(s)sino~sds o o

and

~(w) = ~c(W) - i~8(w).

Substituting (5.4)-(5.5) into (5.3), and integrating over the period of oscillation, up to a positive multiplicative constant we obtain:

Vu2- [(Go + G-~c(W)) T - (Go + G-~c(W))] VUl +

-Vul"G'sVUl - V u 2 " G ' s V l t 2 + (Mo + B~) ' ( t~2VUl - t~l u2) +

1 + lw [ao + a'~c(W)](t ~2 + 022) + - ~ o [gl'(Ko + kc(w))g l +

+gl ' (k s (o ) ) - ] r 2 -t-g2" (Ko + kc(o)))g2] > 0

where T denotes transposition. A strightforward calculation yields

(5.6) Im {(Go + G';(w)) Vu.Vu* + (Mo + B ) 0Vu*} +

1 { 1 } + - - R e [ao+a'(w)]O~*+--(Ko+fe(w))g.g* > 0

o) 0 o

where the superscript * denotes complex conjugate. If we state the fading memory principle in the weak form

(5.7) G', a ' , k e L I ( R + ) A L2(R+),

then (5.6) is well-defined for every w > 0. When k - 0, it is worth nothing that (5.6) is a particular case of inequality (3.9') in

[12]. In particular, under isothermal conditions (0 = 0, g = 0) we recover classical restrictions on the stress-strain relaxation tensor G, namely

(5.8) Go = Go T, G ~ = G T ,

(5.9) Im {(Go + G-~(w))Vu.Vu*} > O, Vw > O,

previously obtained in [5] for viscoelastic solids.

CLAUDI0 GIORGI - MARIA GRAZIA NASO: Exponential stability, etc. 65

Restricting our attention to uniform thermal processes ( V u = O , g = O ) , we have

(5.10) Re {ao + ~ ' (w)} > 0 , Y w > 0 .

On the other hand, if tempera ture is held constant (v ~ = 0) we obtain

(5.11) Re {Ko + k(w)} > 0 , Vw > 0 .

Relations (5.10) and (5.11) recover a previous result relative to rigid conductors with memory (see [6]).

Finally, letting w--> ~ into (5.6), by (5.8)-(5.9) it follows (see [23])

(5.12) Mo = - B

and letting w--*0 + af ter multiplying (5.6) by w, we have

oo

(5.13) Ko + ~k(s) ds >I O . 0

Since

by assumption.

a(0) + ~ ' (0 ) = lim a(s) =0 8--->oo

Acknowledgments. The authors wish to thank Vittorino Pata for helpful discussion and the referee for her/his pert inent comments.

This work has been performed under the auspices of G.N.F.M. - C.N.R. and partial- ly supported by Italian M.U.R.S.T. through the Project ,,Metodi Matematici nella Mec- canica dei Sistemi Continui, .

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