A theory of thermoelasticity with diffusion under Green‐Naghdi models

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ZAMM · Z. Angew. Math. Mech., 1 – 16 (2013) / DOI 10.1002/zamm.201300050

A theory of thermoelasticity with diffusion under Green-Naghdi models1

Moncef Aouadi1,∗, Barbara Lazzari2,∗∗, and Roberta Nibbi2,∗∗∗2

1 Department of Mathematics and Computer Science, Institut Superieur des Sciences Appliquees et de Technologie de3

Mateur, University of Carthage, Tunisia4

2 Department of Mathematics, University of Bologna, 5 Piazza di Porta S. Donato, 40126 Bologna, Italy5

Received 26 February 2013, revised 25 May 2013, accepted 8 June 20136

Published online ♣ 20137

Key words Thermoelastic diffusion, Green-Naghdi theory, well-posedness, asymptotic behavior, localization in time.8

In this paper, we use the Green-Naghdi theory of thermomechanics of continua to derive a nonlinear theory of thermoe-lasticity with diffusion of types II and III. This theory permits propagation of both thermal and diffusion waves at finitespeeds. The equations of the linear theory are also obtained. With the help of the semigroup theory of linear operators weestablish that the linear anisotropic problem is well posed and we study the asymptotic behavior of the solutions. Finally,we investigate the impossibility of the localization in time of solutions.

c© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction9

The usual theory of heat conduction based on the Fourier’s law allows the phenomena of ”infinite diffusion velocity” which10

is not well accepted from a physical point of view. This is referred to as the paradox of heat conduction. It is physically11

unrealistic since it implies that thermal signals propagate with infinite speed. The articles of Dreyer and Struchtrup [1] and12

Caviglia et al. [2] provide an extensive survey of works on experiments involving the propagation of heat as a thermal wave.13

They report instances where the phenomena of second sound has been observed in several kinds of materials. This kind of14

fact has provoked intense activity in the field of heat propagation.15

In contrast to the conventional thermoelasticity, nonclassical theories came into existence during the last two decades.16

These theories, referred to as generalized thermoelasticity, were introduced in the literature in an attempt to eliminate17

the shortcomings of the classical dynamical thermoelasticity. A survey article of representative theories in the range of18

generalized thermoelasticity is due to Hetnarski and Ignaczak [3].19

Green-Naghi [4, 5] developed a thermomechanical theory of deformable continua that relies on an entropy balance law20

rather than an entropy inequality, where the heat conduction does not agree with the usual one (see also [6]). However, we21

want to mention the total compatibility of the entropy balance law with the entropy inequality. They proposed the use of22

the thermal displacement23

α(x, t) =∫ t

t0

θ(x, s)ds + α0,24

where θ is the empirical temperature, and considered three theories labelled as type I, II, and III, respectively. These theories25

were based on an entropy balance law rather than the usual entropy inequality. The type I thermoelasticity coincides with26

the classical one; in type II, the heat is allowed to propagate by means of thermal waves but without dissipating energy27

and, for this reason, it is also known as thermoelasticity without energy dissipation. The heat equation of type III, where28

the heat flux is a combination of type I and II, contains both type I and II as limiting cases. In addition, the thermoelasticity29

of type III allows the constitutive functions for free energy, stress tensor, entropy and heat flux to depend on the strain30

tensor, the time derivative of the thermal displacement, the gradient of thermal displacement and the time derivative of31

the gradient of thermal displacement. This theory allows the dissipation energy, but the heat flux is partially determined32

from the Helmholtz free energy potential. Both, type II and III, overcome the unnatural property of Fourier’s law of infinite33

propagation speed and imply a finite wave propagation.34

All these theories have recently been the subject of great amount of works (as a matter of illustration see [7–14]).35

∗ Corresponding author E-mail: moncef [email protected]∗∗ E-mail: [email protected]∗∗∗ E-mail: [email protected]

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2 M. Aouadi et al.: A theory of thermoelasticity with diffusion under Green-Naghdi models

We may conclude that the Green-Naghdi theory (of heat conduction) is a good model to explain the heat conduction36

for several kinds of solids and fluids. A natural question is to know what happens when the diffusion effect is added to the37

thermal effect. Diffusion can be defined as the random walk of an ensemble of particles from regions of high concentration38

to regions of lower concentration. Thermodiffusion in an elastic solid is due to coupling of the fields of temperature, mass39

diffusion and that of strain. In fact, the development of high technologies in the years before, during, and after the second40

world war pronouncedly affected the investigations in which the fields of temperature and diffusion in solids cannot be41

neglected. At elevated and low temperatures, the processes of heat and mass transfer play a decisive role in many satellites42

problems, returning space vehicles, and landing on water or land. There is now a great deal of interest in the study of43

diffusion, due to its many applications in geophysics and industrial applications. In integrated circuit fabrication, diffusion44

is used to introduce dopants in controlled amounts into the semiconductor substrate. Diffusion is also used to form the base45

and emitter in bipolar transistors, form integrated resistors, form the source/drain regions in MOS transistors, and dope46

polysilicon gates in MOS transistors. In most of these applications, the concentration is calculated using what is known47

as Fick’s law. This is a simple law that does not take into consideration the mutual interaction between the introduced48

substance and the medium into which it is introduced or the effect of the temperature on this interaction. The phenomenon49

of diffusion is used to improve the conditions of oil extractions (seeking ways of more efficiently recovering oil from oil50

deposits). These days, oil companies are interested in the process of diffusion for more efficient extraction of oil from oil51

deposits52

Nowacki [15] developed the classic theory of thermoelastic diffusion of type I. Sherief et al. [16] derived the ther-53

moelastic diffusion theory under Cattaneo’s law. Recently, Aouadi [17–20] derived the general equations of motion and54

constitutive equations of different thermoelastic diffusion theories with some regularity, stability and existence theorems.55

In this paper we extend the theory of thermoelasticity of type II and III to include diffusion effects. We assume that the56

constituents have a common temperature and a common diffusion and that every thermodynamical process that takes place57

is irreversible. This model has not been treated by other authors. We believe that the mathematical and physical analysis58

will reveal the usefulness of this new theory and it is to this end that the present paper is addressed.59

The organization of this paper is as follows. In Sect. 2 we use the theory established by Green-Naghdi [6] to obtain a60

nonlinear theory of thermoelastic diffusion of type II and III, which admits the possibility of ”second sound”. The process61

of linearization of the obtained equations is presented in Sect. 3. With the help of the semigroup theory of linear operators62

an existence result is obtained in Sect. 4. In Sect. 5, the asymptotic behavior for the solutions of type III problem is studied.63

Finally, in Sect. 6, we investigate the impossibility of the localization in time of solutions of type III problem. It is worth64

noting that we focus on the analysis of the qualitative properties of solutions of type III problem. However, some particular65

aspects of the type II problem are also pointed out.66

2 Nonlinear theory67

In this section we present a nonlinear theory of thermoelasticity with diffusion in the context of the Green-Naghdi models68

of type II and III.69

We consider a continuous body that at time τ0 occupies a bounded region Ω of the Euclidean three-dimensional space70

with smooth boundary ∂Ω. We take the configuration Ω as reference configuration and refer the motion of the continuum to71

the reference configuration. Fixed system of rectangular cartesian axes, we denote with Xi the coordinates of a point in the72

reference configuration and with xi the coordinates of the same point at time t, where xi = xi(X1, X2, X3, t). We assume73

that the functions xi are continuously differentiable as it is necessary.74

For any sub-body we denote with B the corresponding region in the reference configuration Ω, which is bounded by a75

regular surface ∂B , and with ni the components of the unit outward normal to ∂B.76

Finally, throughout this paper, we shall employ the usual summation and differentiation conventions: Latin subscripts77

are understood to range over the integers (1, 2, 3), summation over repeated subscripts is implied and subscripts preceded78

by a comma denote partial differentiation with respect to the corresponding cartesian coordinate. Moreover a superposed79

dot denotes the partial time derivative.80

For a thermo-diffusive elastic material, following Nowacki [15], we postulate, for every subregion B of Ω and every81

time t, a mass conservation law which relates the variation of the concentration C to the flux of mass diffusion as follows82 ∫B

Cdv = −∫

∂B

ηda, (2.1)83

where η is the internal diffusive mass flux per unit mass measured per unit area of the surface ∂B.84

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Moreover, following the arguments of Green and Nagdhi [4], we postulate, for every subregion B of Ω and every time85

t, the following balance equations for the energy and the entropy86 ∫B

ρ(xixi + e)dv =∫

B

ρ(fixi + sT )dv +∫

∂B

(tixi − q)da, (2.2)87

88 ∫B

ρSdv =∫

B

ρ(s + ξ)dv −∫

∂B

Φda. (2.3)89

Here ρ denotes the (constant) density in the reference configuration, e is the internal energy per unit mass, fi is the body90

force per unit mass, T is the absolute temperature, s is the external rate of supply of entropy per unit mass, S is the entropy91

per unit mass, ξ is the internal rate of production of entropy per unit mass, while ti is the stress vector, q and Φ are the92

internal flux of heat and entropy, respectively, per unit mass measured per unit area of the surface ∂B.93

By using the invariance property under superposed rigid translations, from Eq. (2.2) we obtain94 ∫B

ρxidv =∫

B

ρfidv +∫

∂B

tida. (2.4)95

Moreover, under suitable hypotheses of regularity, the classical technique of the Cauchy tetrahedron applied to Eqs. (2.1)–96

(2.4) yields97

Φ = Φknk , ti = Tiknk , q = qknk, η = ηknk, (2.5)98

where Φk is the entropy flux vector, Tik is the first Piola Kirchhoff stress tensor, qk is the heat flux vector, and ηk is the flux99

of mass diffusion.100

Therefore, thanks to the arbitrariness of B, we obtain the following local form for the balance equations101

C = −ηk,k,

ρxi = tki,k + ρfi,

ρe = tkixi,k + ρsT − qk,k,

ρS = ρ(s + ξ) − Φk,k.

(2.6)102

If we introduce the specific Helmholtz free energy per unit mass103

Ψ = e − TS, (2.7)104

the above system gives105

ρ(Ψ + T S) = tkixi,k − ρξT − ΦkT,k − (qk − ΦkT ),k. (2.8)106

According to Lebon et al. [21] we assume that the entropy flux, the heat flux and the mass diffusion flux satisfy the following107

relation108

TΦk = qk − Pηk, (2.9)109

where P is the chemical potential. Taking into account (2.6)1 and (2.9), Eq. (2.8) becomes110

ρ(Ψ + T S) = tkixi,k − ρξT − ΦkT,k − ηkP,k + PC. (2.10)111

According to the Green-Nagdhi theory [4], we introduce the thermal displacement α whose derivative coincides with the112

absolute temperature, i.e. α = T . This scalar, on the macroscopic scale, is regarded as representing some ”mean” thermal113

displacement magnitude on the molecular scale. In a similar way, we introduce a scalar function β related to the chemical114

potential by the equation β = P .115

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4 M. Aouadi et al.: A theory of thermoelasticity with diffusion under Green-Naghdi models

2.1 Type II – dissipationless theory116

We assume that the response functions117

Ψ, tkj , S, P, Φk, ηk, ξ (2.11)118

depend on the set of the independent variables119

A1 = (xi,k, T, C, α,k, β,k).120

Thus, we consider constitutive equations of the form121

F = F(A1) (2.12)122

and we assume that the response functions are of C1−class.123

Using the chain rule124

Ψ =∂Ψ

∂xj,kxj,k +

∂Ψ∂T

T +∂Ψ∂C

C +∂Ψ∂α,k

α,k +∂Ψ∂β,k

β,k, (2.13)125

126

the comparison of Eqs. (2.10) and (2.13) yields127 (ρ

∂Ψ∂T

+ ρS

)T +

(ρ

∂Ψ∂xj,k

− tkj

)xj,k +

(ρ

∂Ψ∂α,k

+ Φk

)α,k128

+

(ρ

∂Ψ∂β,k

+ ηk

)β,k +

(ρ

∂Ψ∂C

− P

)C + ρT ξ = 0 (2.14)129

130

which must hold for all choice of T , xj,k, α,k, β,k, and C . From this equality we see that the constitutive equations are131

compatible with the energy equation if satisfy the following relations132

Ψ = Ψ(A1), S = − ∂Ψ(A1)∂T

, tkj = ρ∂Ψ(A1)∂xj,k

,

Φk = −ρ∂Ψ(A1)

∂α,k, ηk = −ρ

∂Ψ(A1)∂β,k

, P = ρ∂Ψ(A1)

∂C, ξ = 0.

(2.15)133

The thermal displacement α and the chemical potential displacement β are defined analogously to the well-known mechan-134

ical displacement. For this, the entropy flux vector Φk and the mass diffusion flux vector ηk are deduced from a potential135

in the same way as the stress tensor is derived in mechanics.136

2.2 Type III – dissipation theory137

Whereas in case of heat flow of type II the response functions (2.11) are assumed to depend on the material deformation138

gradient xi,k, the temperature T , the concentration C, the thermal displacement gradient α,k and the chemical potential139

displacement gradient β,k, for type III we now add the dependency on the temperature gradient α,k and on the chemical140

potential gradient β,k. Hence, we assume that the response functions (2.11) depend on the set of the independent variables141

A2 = (xi,k, T, C, α,k, β,k, α,k, β,k) = (xi,k, T, C, α,k, β,k, T,k, P,k).142

In this case, using the chain rule143

Ψ =∂Ψ

∂xj,kxj,k +

∂Ψ∂T

T +∂Ψ∂C

C +∂Ψ∂α,k

T,k +∂Ψ∂β,k

P,k +∂Ψ∂T,k

T,k +∂Ψ∂P,k

P,k, (2.16)144

the comparison of Eqs. (2.10) and (2.16) yields145

∂Ψ∂T,k

= 0 ,∂Ψ∂P,k

= 0,146

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ZAMM · Z. Angew. Math. Mech. (2013) / www.zamm-journal.org 5

that is Ψ = Ψ(xi,k, T, C, α,k, β,k) = Ψ(A1), and147

S = − ∂Ψ(A1)∂T

, tkj = ρ∂Ψ(A1)∂xj,k

, P = ρ∂Ψ(A1)

∂C,148

149 (ρ

∂Ψ∂α,k

+ Φk

)T,k +

(ρ

∂Ψ∂β,k

+ ηk

)P,k + ρT ξ = 0. (2.17)150

3 Linear theory151

We consider a reference configuration which is in thermal and diffusive equilibrium and free from stresses, with α and β152

constant. We assume that the deformations and the changes of temperature and of concentration are very small with respect153

to the reference configuration in such way that, if C0 and T0 are respectively the (constant) concentration and the (constant)154

absolute temperature of the body in the reference configuration, we can write155

xi − Xi = ui = εu′i, T − T0 = θ = εθ′, C − C0 = γ = εγ′, (3.1)156

where ε is a constant small enough for squares and higher powers to be neglected, and u′i, θ′, and γ′ are independent on ε.157

Under these hypotheses, the strain tensor is approximated with158

eik =12(ui,k + uk,i).159

3.1 Green-Naghdi theory of type II160

The set of the independent variables for the Green-Naghdi model of type II (without energy dissipation) becomes161

A1 = (eik, θ, γ, τ,k, ℘,k),162

where163

τ =∫ t

t0

θds, ℘ =∫ t

t0

Pds.164

To obtain a linear theory, we consider the free energy Ψ function in the quadratic approximation165

ρΨ =12

Aijkleijekl +aijeijθ+bijeijγ− θγ− ρcE

2T0θ2+

12�γ2+

12Kijτ,iτ,j +

12

Hij℘,i℘,j +cijτ,i℘,i, (3.2)166

where ρ denotes the mass density, cE is the specific heat at constant strain, Aijkl is the tensor of elastic constants, aij and bij167

are the tensors of thermal and diffusion expansions, respectively. The constants and � are measures of thermodiffusion168

and diffusive effects, respectively. Kij and Hij are tensors of thermal and diffusion conductivity, respectively. The tensor169

cij is measure of thermodiffusion gradient displacement.170

The compatibility conditions (2.15) give the following linear constitutive equations171

tij = Aijklekl + aijθ + bijγ,

ρS = −aijeij +ρcE

T0θ + γ,

P = bijeij − θ + �γ,

Φi = −Kijτ,j − cij℘,j,

ηi = −Hij℘,j − cijτ,j .

(3.3)172

Moreover, by using (3.3), the linear approximation of (2.9) is given by173

qi = T0Φi = −T0(Kijτ,j + cij℘,j). (3.4)174

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6 M. Aouadi et al.: A theory of thermoelasticity with diffusion under Green-Naghdi models

For the case of isotropic media, we have175

Aijkl = λδijδkl+μδikδjl+μδilδjk, aij = −β1δij , bij = −β2δij , Kij = kδij , cij = �δij , Hij = hδij ,

(3.5)176

where δij is the Kronecker’s delta, λ and μ are Lame’s constants, β1 = (3λ + 2μ)αt and β2 = (3λ + 2μ)αc, αt is the177

coefficient of linear thermal expansion and αc is the coefficient of linear diffusion expansion.178

If the material is isotropic, then the constitutive equations become179

tij = λekkδij + 2μeij − β1θδij − β2γδij ,

ρS = β1ekk +ρcE

T0θ + γ,

P = −β2ekk − θ + �γ,

Φi = −kτ,i − �℘,i

ηi = −h℘,i − �τ,i,

qi = −T0(kτ,j + �℘,j).

180

If we will use the chemical potential as a state variable instead of the concentration, we can give the following formula-181

tion of the constitutive equations (3.3) and (3.4) in the linear theory:182

tij = αijklekl + γijθ + βijP,

ρS = −γijeij + cθ + κP,

γ = −βijeij + κθ + mP,

Φi = −Kijτ,j − cij℘,j,

ηi = −Hij℘,j − cijτ,j ,

qi = −T0(Kijτ,j + cij℘,j),

(3.6)183

where184

αijkl = Aijkl − 1�

bijbkl, γij = aij +

�bij , βij =

1�

bij , c =ρ0cE

T0+

2

�, κ =

�, m =

1�

. (3.7)185

In the sequel we assume that the above constitutive coefficients satisfy the following symmetry relations186

αijkl = αklij , cij = cji, αij = αji, βij = βji, Kij = Kji, Hij = Hji. (3.8)187

By introducing the constitutive equations (3.6) into the balance equations (2.6), we obtain the following evolutive equations188

of the theory of thermoelastic diffusion materials of type II (without energy dissipation)189

ρui = (αjiklekl + γjiθ + βjiP ),j + ρfi,

cτ = γij eij + (Kijτ,j + cij℘,j),i − κ℘ + ρs,

m℘ = βij eij + (Hij℘,j + cijτ,j),i − κτ .

(3.9)190

3.2 Green-Naghdi theory of type III191

By using the same hypotheses and notations of the previous subsection, the set of the independent variables for the Green-192

Naghdi model of type III (with energy dissipation) becomes193

A2 = (eik, θ, γ, τ,k, ℘,k, θ,k, P,k)194

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and, as a consequence of (2.17), the quadratic approximation of the free energy Ψ is given by (3.2). Moreover, the consti-195

tutive equations for tkj , S and P are equal to (3.3)1,2,3, while the condition (2.17) leads to196 (ρ

∂Ψ∂τ,k

+ Φk

)θ,k +

(ρ

∂Ψ∂℘,k

+ ηk

)P,k + ρT0ξ = 0. (3.10)197

In the context of a linear theory, the condition (3.10) is satisfied if we choose198

Φi = −[Kijτ,j + cij℘,j + Kijθ,j + cijP,j

],

ηi = −[Hij℘,j + cijτ,j + HijP,j + cijθ,j

],

qi = T0Φi = −T0

[Kijτ,j + cij℘,j + Kijθ,j + cijP,j

],

ρξ = HijP,iP,j + Kijθ,iθ,j + 2cijP,jθ,i,

(3.11)199

where Hij , Kij , and cij are tensors characteristic of this theory.200

If the material is isotropic, then the constitutive equations become201

tij = λekkδij + 2μeij − β1θδij − β2γδij ,

ρS = β1ekk +ρcE

T0θ + γ,

P = −β2ekk − θ + �γ,

Φi = − [k1τ,i + �1℘,i + k2θ,i + �2P,i

],

ηi = − [h1℘,i + �1τ,i + h2P,i + �2θ,i,],

qi = −T0

[k1τ,i + �1℘,i + k2θ,i + �2P,i

].

202

Using the chemical potential as a state variable instead of the concentration, we obtain the following evolutive equations203

of the theory of thermoelastic diffusion materials of type III204

ρui = (αjiklekl + γjiθ + βjiP ),j + ρfi,

cτ = γij eij + (Kijτ,j + cij℘,j + Kijθ,j + cijP,j),i − κ℘ + ρs,

m℘ = βij eij + (Hij℘,j + cijτ,j + HijP,j + cijθ,j),i − κτ ,

(3.12)205

where the above constitutive coefficients are given by Eq. (3.7) and satisfy the following symmetry relations206

αijkl = αklij , cij = cji, αij = αji, βij = βji,

Kij = Kji, Hij = Hji, Kij = Kji, Hij = Hji cij = cji.(3.13)207

Remark that the evolutive equations (3.9) of the thermoelastic diffusion theory of type II (without energy dissipation) can208

be deduced from the evolutive equations (3.12) of the thermoelastic diffusion theory of type III (with energy dissipation)209

by taking Kij = Hij = cij = 0.210

To the field of equations of type II and III we adjoin boundary and initial conditions. The components of the surface211

traction, the heat flux and the diffusion flux at regular points of ∂Ω, are respectively given by212

fi = tjinj, q = qini, η = ηini .213

Summarizing, the following initial boundary value problems are to be solved:214

(i) Type II problem : Find (ui, vi, τ, θ, ℘, P ) solution of (3.9) subject to the initial conditions215

ui(·, 0) = u0i , vi(·, 0) = v0

i , τ(·, 0) = τ0,

θ(·, 0) = θ0, ℘(·, 0) = ℘0, P (·, 0) = P 0 in Ω,(3.14)216

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8 M. Aouadi et al.: A theory of thermoelasticity with diffusion under Green-Naghdi models

and the boundary conditions217

ui = ui on ∂Ωu × (0,∞), τ = τ on ∂Ωτ × (0,∞), ℘ = ℘ on ∂Ω℘ × (0,∞),

tjinj = fi on ∂Ωσ × (0,∞), qini = q on ∂Ωq × (0,∞), ηini = η on ∂Ωη × (0,∞),(3.15)218

where ui, τ , ℘, fi, q, and η are prescribed functions, u0i , v0

i , τ0, θ0, ℘0, and P 0 are given and219

∂Ω = ∂Ωu ∪ ∂Ωσ = ∂Ωτ ∪ ∂Ωq = ∂Ω℘ ∪ ∂Ωη, and ∂Ωu ∩ ∂Ωσ = ∂Ωτ ∩ ∂Ωq = ∂Ω℘ ∩ ∂Ωη = ∅ .220

(ii) Type III problem : Find (ui, vi, τ, θ, ℘, P ) solution of (3.12) subject to the initial conditions (3.14) and the boundary221

conditions (3.15).222

In the following, we assume that223

Δ = cm − κ2 > 0. (3.16)224

Note that this condition implies that225

cθ2 + 2κθP + mP 2 > 0.226

Condition (3.16) is needed to stabilize the thermoelastic diffusion system (see [16–22] for more information on this).227

Some qualitative properties of the solutions of type III problem are studied in the next. However, some particular aspects228

of the type II problem are also pointed out.229

4 Well-posedness of the linear system230

We will prove the existence, uniqueness and continuous dependence from the initial values and the external loads of the231

solution for system (3.12) using semi-group theory. Seeking for simplicity, we will restrict ourselves to homogeneous232

boundary conditions233

u = 0, τ = 0, ℘ = 0 on ∂Ω × (0,∞), (4.1)234

where u denotes the vector of components ui, i.e. u = (ui).235

In the rest of the paper we assume:236

(i) relations (3.13) and (3.16) are satisfied;237

(ii) ρ > 0 and1238

HijP,iP,j + Kijθ,iθ,j + 2cijP,jθ,i ≥ 0 ; (4.2)239

(iii) there exists a positive constant c0 such that240 ∫Ω

(αijklekleij + 2cijτ,i℘,j + Kijτ,iτ,j + Hij℘,i℘,j

)dv ≥ c0

∫Ω

(eijeij + τ,iτ,i + ℘,i℘,i)dv. (4.3)241

We now wish to transform the boundary-initial-value problem defined by Eqs. (3.12), the boundary conditions (4.1) and the242

initial conditions (3.14) to an abstract problem on a suitable Hilbert space. In what follows we use the notation v = u, θ =243

τ , P = ℘. Let244

£ = {(u, v, τ, θ, ℘, P ); u ∈ W1,20 (Ω); v ∈ L2(Ω); τ, ℘ ∈ W 1,2

0 (Ω); θ, P ∈ L2(Ω)},245

where W 1,20 (Ω), L2(Ω) are the familiar Sobolev spaces and246

W1,20 (Ω) = [W 1,2

0 (Ω)]3, L2(Ω) = [L2(Ω)]3.247

We introduce the inner product in £ defined by248

〈(u, v, τ, θ, ℘, P ), (u∗, v∗, τ∗, θ∗, ℘∗, P ∗)〉

=12

∫Ω

(ρvivi

∗ + cθθ∗ + κ(θP ∗ + θ∗P ) + mPP ∗ + 2W [(u, τ, ℘), (u∗, τ∗, ℘∗)])dv,

(4.4)249

1 The inequality sign is a consequence of the Second Law of Thermodynamics, which requires the non-negativeness of the functional ξ (see [4]) andof our choice (3.11)4.

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ZAMM · Z. Angew. Math. Mech. (2013) / www.zamm-journal.org 9

where250

2W [(u, τ, ℘), (u∗, τ∗, ℘∗)] = αijklekle∗ij + cij(τ∗

,i℘,j + τ,j℘∗,i) + Kijτ,iτ

∗,j + Hij℘,i℘

∗,j.251

If we recall the assumption (4.3), the first Korn inequality and the Poincare inequality, we conclude that252 ∫Ω

W [(u, τ, ℘), (u, τ, ℘)]dv253

defines a norm that is equivalent to the usual norm in W1,20 (Ω) × W 1,2

0 (Ω) × W 1,20 (Ω). Hence, the bilinear form (4.4)254

defines an inner product equivalent to the usual one in £.255

We introduce the following operators256

Aiu = ρ−1(αjikluk,l),j ,

Biθ = ρ−1(γjiθ),j ,

CiP = ρ−1(βjiP ),j ,

Dv =mγij − κβij

cm − κ2vi,j ,

Eτ =1

cm − κ2((mKij − κcij)τ,j),i,

Gθ =1

cm − κ2((mKij − κcij)θ,j),i,

J℘ =1

cm − κ2((mcij − κHij)℘,j),i,

IP =1

cm − κ2((mcij − κHij)P,j),i,

Lv =1

cm − κ2((cβij − κγij)vj),i,

Mτ =1

cm − κ2((ccij − κKij)τ,j),i,

Nθ =1

cm − κ2((ccij − κKij)θ,j),i,

Q℘ =1

cm − κ2((cHij − κcij)℘,j),i,

RP =1

cm − κ2((cHij − κcij)P,j),i,

(4.5)257

and consider the matrix operator A on £ defined by258

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 Id 0 0 0 0A 0 0 B 0 C0 0 0 Id 0 00 D E G J I

0 0 0 0 0 Id

0 L M N Q R

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠(4.6)259

with domain260

D = D(A) = {(u, v, τ, θ, ℘, P )T ∈ £; A(u, v, τ, θ, ℘, P )T ∈ £},261

where Id and Id are the identity operators in the respective spaces, A = (Ai), B = (Bi), C = (Ci), and AT is the transpose262

of A.263

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10 M. Aouadi et al.: A theory of thermoelasticity with diffusion under Green-Naghdi models

In the frame of type II theory, we have that Kij = 0, cij = 0, and Hij = 0, so that G = I = N = R = 0 in Eqs. (4.5)264

and (4.6).265

It is clear that266

(W1,20 (Ω) ∩ W2,2(Ω)) × W1,2

0 (Ω) × (W 1,20 (Ω) ∩ W 2,2(Ω)) × W 1,2

0 (Ω)267

268

×(W 1,20 (Ω) ∩ W 2,2(Ω)) × W 1,2

0 (Ω)269

is a subset of D which is dense in £.270

The boundary-initial value problem (3.12), (4.1), and (3.14) can be transformed into the following equation in the Hilbert271

space £,272

dU(t)dt

= AU(t) + F(t), U(0) = U0, (4.7)273

where274

U = (u, v, τ, θ, ℘, P ), F = (0, fi, 0,mρ

cm − κ2s, 0,

−κρ

cm − κ2s), U0 = (u0, v0, τ0, θ0, ℘0, P 0).275

Now, we use the theory of semigroups of linear operators to obtain the existence of solutions for the Eq. (4.7).276

Lemma 4.1. The operator A satisfies the inequality277

< AU ,U >≤ 0278

for every U ∈ D(A), solution of (4.7).279

P r o o f. Let U = (u, v, τ, θ, ℘, P ) ∈ D(A). Using the divergence theorem and the boundary conditions, we have280

< AU ,U > =∫

Ω

[W((u, τ, ℘), (u, τ, ℘)

)− vj,i(tji − γjiθ − βjiP ) − Φiθ,i − ηiP,i

]dv

= −∫

Ω

(Kijθ,iθ,j + HijP,iP,j + 2cijP,jθ,i)dv .281

The thesis follows from our hypothesis (4.2).282

In the context of type II theory (Kij = cij = Hij = 0), this lemma implies < AU ,U >= 0, which means conservation283

of the energy284

E(t) =12

∫Ω

(ρvivi + cθ2 + 2κθP + mP 2 + αijklekleij + 2cijτ,i℘,j + Kijτ,iτ,j + Hij℘,i℘,j

)dv. (4.8)285

It is worth remarking that this quantity is also conserved even if we do not impose conditions (i) − (iii).286

Lemma 4.2. The operator A has the property that287

Range(I − A) = £.288

P r o o f. Let U∗ = (u∗, v∗, τ∗, θ∗, ℘∗, P ∗) ∈ £. We must prove that the equation289

U −AU = U∗290

has a solution U = (u, v, τ, θ, ℘, P ) ∈ D. This equation leads to the system291

u∗ = u − v,

τ∗ = τ − θ,

℘∗ = ℘ − P,

v∗ = v − (Au + Bθ + CP ),

θ∗ = θ − (Dv + Eτ + Gθ + J℘ + IP ),

P ∗ = P − (Lv + Mτ + Nθ + Q℘ + RP ).

(4.9)292

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ZAMM · Z. Angew. Math. Mech. (2013) / www.zamm-journal.org 11

Substituting the first three equations in the others, we obtain the following system with unknowns (u, τ, ℘)293

u∗ + v∗ − Bτ∗ − C℘∗ = u − (Au + Bτ + C℘),

θ∗ + τ∗ − Du∗ − Gτ∗ − I℘∗ = τ − (Du + (E + G)τ + (I + J)℘),

P ∗ + ℘∗ − Lu∗ − Nτ∗ − R℘∗ = ℘ − (Lu + (M + N)τ + (Q + R)℘).

(4.10)294

To solve this system, we introduce the following bilinear form on W1,20 (Ω) × W 1,2

0 (Ω) × W 1,20 (Ω),295

Λ[Γ1, Γ2] =< Σ, Υ >, (4.11)296

where297

Γ1 = (u, τ, ℘), Γ2 = (u, τ , ℘), Σ = (u(1), τ (1), ℘(1)), Υ = (ρu, cτ , m℘)298

and299

u(1) = u − (Au + Bτ + C℘),

τ (1) = τ − (Du + (E + G)τ + (I + J)℘),

℘(1) = ℘ − (Lu + (M + N)τ + (Q + R)℘).

300

A direct calculation shows that Λ is bounded in each variable. Using the divergence theorem, we have301

Λ[Γ1, Γ1] =∫

Ω

[ρuiui+cτ2+2κτ℘+m℘2+Kijθ,iθ,j+HijP,iP,j+2cijθ,iP,j+2W [(u, τ, ℘), (u, τ, ℘)]

]dv.302

In view of our assumptions on the constitutive coefficients, we see that Λ is coercive on W−1,2(Ω)×W−1,2(Ω)×W−1,2(Ω).303

On the other hand, it is easy to see that the vector304

(u∗ + v∗ − Bτ∗ − C℘∗, θ∗ + τ∗ − Du∗ − Gτ∗ − I℘∗, P ∗ + ℘∗ − Lu∗ − Nτ∗ − R℘∗)305

lies in W−1,2(Ω) × W−1,2(Ω) × W−1,2(Ω). Hence the Lax-Milgram theorem [22] implies the existence of (u, τ, ℘) ∈306

W1,20 (Ω) × W 1,2

0 (Ω) × W 1,20 (Ω) which solves Eq. (4.10). Now, we may also conclude the existence of v ∈ W1,2

0 (Ω),307

θ ∈ W 1,20 (Ω) and P ∈ W 1,2

0 (Ω) solving system (4.9).308

The previous lemmas lead to next theorem.309

Theorem 4.1. The operator A generates a semigroup of contraction in £.310

P r o o f. The proof follows from Lumer-Phillips corollary to the Hille-Yosida theorem [23].311

It is worth remarking that this theorem implies that the dynamical system generated by the equations of thermoelasticity312

with diffusion of type III (or type II) are stable in the sense of Lyapunov.313

Theorem 4.2. Assume that fi, s ∈ C1([0,∞), L2) and U0 is in the domain of the operator A. Then, there exists a314

unique solution U(t) ∈ C1([0,∞),£) ∩ C0([0,∞),D(A)) to the problem (4.7).315

Since the solutions are defined by means of a semigroup of contraction, we have the estimate316

‖U(t)‖ ≤ ‖U0‖£ +∫ t

0

(‖fi(ξ)‖L2 + ‖s(ξ)‖L2

)dξ317

which proves the continuous dependence of the solutions upon initial data and body loads. Thus, under assumptions (i) −318

(iii) the problem of linear thermoelasticity with diffusion of type III (or type II) is well posed.319

5 Asymptotic behavior of solutions320

In this section we study the asymptotic behavior of solutions, whose existence has been proved previously, in the homoge-321

nous case (fi=0, s=0). In particular we are interested in the relation between dissipation effects and time decay of solutions.322

Therefore, we will continue to assume that the assumptions (i) − −(iii) considered in the previous section hold. However323

it is worth noting that the results for this section only hold for type III theory.324

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12 M. Aouadi et al.: A theory of thermoelasticity with diffusion under Green-Naghdi models

To this end, we recall that for a semigroup of contraction, the precompact orbits tend to the ω−limit sets if its generator325

A has only the fixed point 0 (see [24]) and the structure of the ω−limit sets is determined by the eigenvectors of eigenvalue326

iλ (where λ is a real number) in the closed subspace327

L =� {U ∈ £; < AU ,U >= 0} ,328

where � C denotes the closed vectorial subspace generated by the set C.329

From the assumptions (i)-(iii) it is easy to check that A−1(0) = 0, while the precompactness of the orbits starting in D330

is a consequence of the following Lemma [23].331

Lemma 5.1. The operator (I − A)−1 is compact.332

P r o o f. Let (un, vn, τn, θn, ℘n, Pn) be a bounded sequence in £ and let (un, vn, τn, θn, ℘n, Pn) be the sequence of333

the respective solutions of the system (4.9). We have334

Λ[Γn, Γn] =< Σn, Υn >, (5.1)335

where336

Γn = (un, τn, ℘n), Σn = (u(1)n , τ (1)

n , ℘(1)n ), Υn = (ρun, cτn, m℘n)337

and338

u(1)n = un − (Aun + Bτn + C℘n),

τ(1)n = τn − (Dun + (E + G)τn + (I + J)℘n),

℘(1)n = ℘n − (Lun + (M + N)τn + (Q + R)℘n).

(5.2)339

In view of the definition of (u(1)n , τ

(1)n , ℘

(1)n ), it follows that it is a bounded sequence in L2(Ω)×L2(Ω)×L2(Ω) and then the340

sequence (un, τn, ℘n) is a bounded sequence in W1,20 (Ω)× W 1,2

0 (Ω)×W 1,20 (Ω). The theorem of Rellich-Kondrasov [25]341

implies that there is exists a subsequence converging in L2(Ω) × L2(Ω) × L2(Ω). In a similar way342

vnj = unj − unj , θnj = τnj − τnj , Pnj = ℘nj − ℘nj343

has a sub-sequence converging in L2(Ω) × L2(Ω) × L2(Ω). Thus we conclude the existence of a sub-sequence344

(unjk, vnjk

, τnjk, θnjk

, ℘njk, Pnjk

)345

which converges in £.346

Now, we can state a theorem on the asymptotic behavior of solutions347

Theorem 5.2. Let U0 = (u0,v0, τ0, θ0, ℘0, P 0) ∈ D(A) and U(t) be the solution of the boundary-initial-value prob-348

lem (4.7) with F = 0. Then349

τ(t), ℘(t) → 0 as t → ∞ in W 1,20 (Ω) and θ(t), P (t) → 0 as t → ∞ in L2(Ω).350

Moreover351

u(t) → 0 as t → ∞ in W1,20 (Ω) and v(t) → 0 as t → ∞ in L2(Ω)352

whenever the system353

Au + λ2u = 0 in Ω,

Du = 0 in Ω,

Lu = 0 in Ω,

u = 0 on ∂Ω

(5.3)354

has only the null solution.355

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ZAMM · Z. Angew. Math. Mech. (2013) / www.zamm-journal.org 13

P r o o f. To prove the theorem we have to study the structure of the ω−limit set. Thus, we must study the equation356

AU = iλU (5.4)357

for some real number λ, where U ∈ D(A) and A = A|L is the generator of a group on L. If U ∈ L then < AU ,U >= 0.358

Under the assumption that the tensors Kij and Hij are definite positive, it follows that θ = P = 0 and then τ = ℘ = 0.359

Thus, the asymptotic behavior of the temperature and the chemical potential is proved.360

Now, Eq. (5.4) can be rewritten as361 ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 Id 0 0 0 0A 0 0 B 0 C0 0 0 Id 0 00 D E G J I

0 0 0 0 0 Id

0 L M N Q R

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

uv0000

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠= iλ

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

uv0000

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠.362

Introducing the first equation into the others we obtain the system (5.3).363

If the system (5.3) has only the trivial solution, we obtain ω−limit(U0) = 0 and u(t) → 0 in W1,20 (Ω) and v(t) →364

0 in L2(Ω) when t → ∞.365

Even if the system (5.3) depends on a lot of parameters, it is the same one considered by Dafermos [24] in the context366

of the classical thermoelasticity. Therefore, we can conclude that, generally, the null solution is the unique solution and we367

expect asymptotic stability; however, it could happen that for certain geometries and choice of the parameters there exists368

nontrivial solutions which lead to undamped solutions of (4.7).369

6 Impossibility of localization in time370

In previous sections we have proved that the solutions of type III theory are stable asymptotically and in the sense of371

Lyapunov. A natural question is to ask if the decay is fast enough to guarantee that the solution vanishes in a finite time.372

In fact, when the dissipation mechanism in a system is sufficiently strong, the localization of solutions in the time variable373

can hold. This means that the decay of the solutions is sufficiently fast to guarantee that they vanish after a finite time.374

In the context of Green-Nagdi thermoelasticity of type III, Quintanilla [11] has shown that the thermal dissipation is not375

strong enough to obtain the localization in time of the solutions. In this section assume the quadratic form (4.2) positive376

definite and prove that the further dissipation effects due to the diffusion are not sufficiently strong to guarantee that the377

thermomechanical deformations vanish after a finite interval of time. This means that, in absence of sources, the only378

solution for the evolutive problem that vanishes after a finite time is the null solution, that is the following theorem holds.379

Theorem 6.1. Let (ui, τ, ℘) be a solution of the system (3.12), (4.1), and (3.14) which vanishes after a finite time t0.380

Then (ui, τ, ℘) ≡ (0.0, 0) for every t ≥ 0.381

In order to prove this theorem, generalizing the technique used in [11], we show the uniqueness of solutions for the382

related backward in time problem.383

Backward in time problems are relevant from the mechanical point of view when we want to have some information384

about what happened in the past by means of the information that we have at this moment.385

For our model, the system of equations which govern the backward in time problem is given by386

ρui = (αjikluk,l − γji τ − βji℘),j + ρfi,

cτ = −γij ui,j + (Kijτ,j + cij℘,j − Kij τ,j − cij ℘,j),i − κ℘ + ρs,

m℘ = −βij ui,j + (Hij℘,j + cijτ,j − Hij ℘,j − cij τ,j),i − κτ .

(6.1)387

Proposition 6.1 (Uniqueness). Let (ui, τ, ℘) be a solution of the system (6.1), (4.1) with null initial data and sources.388

Then (ui, τ, ℘) = (0, 0, 0) for every t ≥ 0.389

P r o o f. Let us introduce the following functionals390

E1(t) =12

∫Ω

(ρuiui + cτ2 + 2κτ℘ + m℘2 + αijklui,juk,l + 2cijτ,i℘,j + Kijτ,iτ,j + Hij℘,i℘,j

)dv ,391

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14 M. Aouadi et al.: A theory of thermoelasticity with diffusion under Green-Naghdi models

E2(t) =12

∫Ω

(ρuiui − cτ2 − 2κτ℘ − m℘2 + αijklui,juk,l − 2cijτ,i℘,j − Kijτ,iτ,j − Hij℘,i℘,j

)dv ,392

E3(t) =∫

Ω

(ρuiui − cττ − m℘℘ − κ(τ℘ + τ℘) +

12Kijτ,iτ,j +

12Hij℘,i℘,j + cijτ,i℘,j

)dv393

and compute their time derivatives.394

By multiplying the first equation of (6.1) by ui, the second one by τ and the third one by ℘, we get395

E1(t) =∫

Ω

(Kij τ,iτ,j + Hij ℘,i℘,j + 2cij τ,j℘,i

)dv.396

On the other hand, if we multiply equation of (6.1) by ui, the second one by −τ and the third one by −℘, we obtain397

E2(t) =∫

Ω

(2γij τ,j ui + 2βij℘,jui − Kij τ,iτ,j − Hij℘,i℘,j − 2cij τ,j℘,i

)dv,398

and, finally, if we multiply the first equation of (6.1) by −ui, the second one by τ and the third one by ℘, we have399

E3(t) = −∫

Ω

(αijklui,juk,l + γijτ,j ui + βij℘,j ui − γij τui,j − βij℘ui,j − 2cijτ,i℘,j400

− Kijτ,iτ,j − Hij℘,i℘,j + cτ2 + 2κτ℘ + m℘2 − ρuiui

)dv .401

402

Moreover, a well-known identity for type III thermoelasticity (see Eq. (3.9) in [10]) for our model becomes403 ∫Ω

(αijklui,juk,l + cτ2 + 2κτ℘ + m℘2

)dv =

∫Ω

(ρuiui + 2cijτ,i℘,j + Kijτ,iτ,j + Hij℘,i℘,j

)dv. (6.2)404

Then we have405

E2(t) =∫

Ω

(αijklui,juk,l − 2cijτ,i℘,j − Kijτ,iτ,j − Hij℘,i℘,j

)dv406

and407

E3(t) =∫

Ω

(γijτ,j ui + βij℘,jui − γij τui,j − βij ℘ui,j

)dv.408

We consider the function409

E(t) =∫ t

0

[εE1(s) + E2(s) + λE3(s)]ds410

=12

∫ t

0

∫Ω

[ερuiui + εcτ2 + 2εκτ ℘ + εm℘2 + (ε + 2)αijklui,juk,l]dv ds411

+12

∫ t

0

∫Ω

{[λKij + (ε − 2)Kij ]τ,iτ,j + 2[λcij + (ε − 2)cij ]τ,i℘,j412

+ [λHij + (ε − 2)Hij ]℘,i℘,j}dv ds + λ

∫ t

0

∫Ω

[ρuiui − cτ τ − m℘℘ − κ(τ℘ + τ℘)]dv ds,413

414

where ε and λ are positive suitable constants such that the quadratic form415 ∫Ω

{[λKij + (ε − 2)Kij ]τ,iτ,j + 2[λcij + (ε − 2)cij ]τ,i℘,j + [λHij + (ε − 2)Hij ]℘,i℘,j} dv416

is positive definite.417

By using the null initial data hypothesis and the Poincare inequality we have418

λ

∫ t

0

∫Ω

[ρuiui − cτ τ − m℘℘ − κ(τ℘ + τ℘)]dv ds ≤ ε

4

∫ t

0

∫Ω

[ρuiui + cτ2 + 2κτ℘ + m℘2] dv ds419

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ZAMM · Z. Angew. Math. Mech. (2013) / www.zamm-journal.org 15

for any t ≤ t0, where t0 is a positive time which depends on λ, ε and the constitutive coefficients. Therefore E(t) is a420

positive definite quadratic form for 0 ≤ t ≤ t0, in particular421

E(t) ≥ 14

∫ t

0

∫Ω

[ερuiui + εcτ2 + 2εκτ ℘ + εm℘2 + (ε + 2)αijklui,juk,l]dv ds (6.3)422

+14

∫ t

0

∫Ω

{[λKij + (ε − 2)Kij]τ,iτ,j + 2[λcij + (ε − 2)cij ]τ,i℘,j + [λHij + (ε − 2)Hij ]℘,i℘,j}dv ds.423

Moreover, recalling the null initial data assumption, we have424

E(t) = (ε − 1)∫ t

0

∫Ω

(Kij τ,iτ,j + Hij ℘,i℘,j + 2cij τ,j℘,i

)dv ds425

+∫ t

0

∫Ω

[2(γij τ,j ui + βij℘,j ui) + λ(γijτ,j ui + βij℘,j ui − γij τui,j − βij℘ui,j)] dv ds.426

427

Choosing 0 < ε < 1 and using the inequality of arithmetic and geometric means, we have428 ∣∣∣∣∣∫ t

0

∫Ω

[2(γij τ,j ui + βij ℘,jui) + λ(γijτ,j ui + βij℘,jui − γij τui,j − βij℘ui,j)] dv ds

∣∣∣∣∣429

≤ (1 − ε)∫ t

0

∫Ω

(Kij τ,iτ,j + Hij ℘,i℘,j + 2cij τ,j℘,i

)dv ds430

+ K1

∫ t

0

∫Ω

ρuiui dv ds + K2

∫ t

0

∫Ω

[cτ2 + 2εκτ ℘ + εm℘2]dv ds + K3

∫ t

0

∫Ω

αijklui,juk,ldv ds431

+12

∫ t

0

∫Ω

{[λKij + (ε − 2)Kij]τ,iτ,j + 2[λcij + (ε − 2)cij ]τ,i℘,j + [λHij + (ε − 2)Hij ]℘,i℘,j

}dv ds,432

433

where the positive constants Ki can be calculated by standard methods, so that434

E(t) ≤ K

∫ t

0

∫Ω

[ερuiui + εcτ2 + 2εκτ ℘ + εm℘2 + (ε + 2)αijklui,juk,l]dv ds (6.4)435

+ K

∫ t

0

∫Ω

{[λKij + (ε − 2)Kij ]τ,iτ,j + 2[λcij + (ε − 2)cij ]τ,i℘,j + [λHij + (ε − 2)Hij ]℘,i℘,j}dv ds436437

with K = max{ 12 , K1, K2, K3}. Inequalities (6.3) and (6.4) yield438

E(t) ≤ 4KE(t) , 0 ≤ t ≤ t0 .439

This inequality and the null initial data imply E(t) ≡ 0 if 0 ≤ t ≤ t0. Reiterating this argument on each subinterval440

[(n − 1)t0, nt0] we obtain E(t) ≡ 0 for t ≥ 0.441

If we take into account the definition of E(t), the uniqueness result is proved.442

7 Conclusions443

The results established in this paper can be summarized as follows:444

(i) We have derived the nonlinear and linear theories of thermoelasticity with diffusion of type II and III in the frame445

of Green-Naghdi thoery. The filed equations of the linear theory are also presented for both isotropic and anisotropic446

solids. We have shown that the type II theory is conservative and the solutions cannot decay with respect to time. It is447

well known that, in general, the solutions of type III decay with respect to time [13].448

(ii) We have proved that the problem of linear thermoelasticity with diffusion of type III (or type II) is well posed. This449

result proves that in the motion following any sufficiently small change in the external system, the solution of the450

initial-boundary value problem is everywhere arbitrary small in magnitude.451

(iii) We have have shown the asymptotic behaviour of solutions of type III theory.452

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16 M. Aouadi et al.: A theory of thermoelasticity with diffusion under Green-Naghdi models

(iv) We have derived the uniqueness of solutions of type III theory for the backward in time problem. Thus, it says the453

impossibility of localization in time of the solutions. From a thermomechanical point of view, this result says that454

combination of thermal and diffusion dissipations in an elastic material is not sufficiently strong to guarantee that the455

thermomechanical deformations vanish after a finite interval of time.456

We believe that by adapting the same analysis, we can prove the impossibility of localization of solutions in the case457

of exterior domains, even when the solutions can be unbounded, whether the spatial variable goes to infinity. This proof is458

omitted for the sake of brevity.459

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