Research Article Exponential Stability in Hyperbolic Thermoelastic Diffusion Problem with Second Sound

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1 International Journal of Differential Equations Volume, Article ID 74843, pages oi:.55//74843 Research Article Exponential Stability in Hyperbolic Thermoelastic Diffusion Problem with Secon Soun Moncef Aouai Département e Mathématique, Institut Supérieur es Sciences Appliquées et e Technologie e Mateur, Route e Tabarka, Mateur 73, Tunisia Corresponence shoul be aresse to Moncef Aouai, moncef aouai@yahoo.fr Receive 4 May ; Accepte 8 June Acaemic Eitor: Sabri Arik Copyright q Moncef Aouai. This is an open access article istribute uner the Creative Commons Attribution License, which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite. We consier a thermoelastic iffusion problem in one space imension with secon soun. The thermal an iffusion isturbancesare moele by Cattaneo s law for heat an iffusion equations to remove the physical paraox of infinite propagation spee in the classical theory within Fourier s law. The system of equations in this case is a coupling of three hyperbolic equations. It poses some new analytical an mathematical ifficulties. The exponential stability of the slightly ampe an totally hyperbolic system is prove. Comparison with classical theory is given.. Introuction The classical moel for the propagation of heat turns into the well-known Fourier s law q k θ,. where θ is temperature ifference to a fixe constant reference temperature, q is the heat conuction vector, an k is the coefficient of thermal conuctivity. The moel using classic Fourier s law inhibits the physical paraox of infinite propagation spee of signals. To eliminate this paraox, a generalize thermoelasticity theory has been evelope subsequently. The evelopment of this theory was accelerate by the avent of the secon soun effects observe experimentally in materials at a very low temperature. In heat transfer problems involving very short time intervals an/or very high heat fluxes, it has been reveale that the inclusion of the secon soun effects to the original theory yiels results which are realistic an very much ifferent from those obtaine with classic Fourier s law.

2 International Journal of Differential Equations The first theory was evelope by Lor an Shulman. In this theory, a moifie law of heat conuction, the Cattaneo s law, τ q t q k θ. replaces the classic Fourier s law. The heat equation associate with this a hyperbolic one an, hence, automatically eliminates the paraox of infinite spees. The positive parameter τ is the relaxation time escribing the time lag in the response of the heat flux to a graient in the temperature. The evelopment of high technologies in the years before, uring, an after the secon worl war pronouncely affecte the investigations in which the fiels of temperature an iffusion in solis cannot be neglecte. The problems connecte with the iffusion of matter in thermoelastic boies an the interaction of mechanoiffusion processes have become the subject of research by many authors. At elevate an low temperatures, the processes of heat an mass transfer play a ecisive role in many satellite problems, returning space vehicles, an laning on water or lan. These ays, oil companies are intereste in the process of thermoiffusion for more efficient extraction of oil from oil eposits. Nowacki evelope the classic theory of thermoelastic iffusion uner Fourier s law. Sherief et al. 3 erive the theory of thermoelastic iffusion with secon soun. Aouai 4 erive the theory of micropolar thermoelastic iffusion uner Cattaneo s law. Recently, Aouai 5 7 erive the general equations of motion an constitutive equations of the linear thermoelastic iffusion theory in the context of ifferent meia, with uniqueness an existence theorems. In recent years, a relevant task has been evelope to obtain exponential stability of solutions in thermoelastic theories. The classical theory was first consiere by Dafermos 8 an Slemro 9 an it has been stuie in the book of Jiang an Racke an the contribution of Lebeau an Zuazua. One shoul mention a paper by Sherief, where the stability of the null solution also in higher imension is prove. We mention also the work of Tarabek 3 who stuie even one imensional nonlinear systems an obtaine the strong convergence of erivatives of solutions to zero. Racke 4 prove the exponential ecay of linear an nonlinear thermoelastic systems with secon soun in one imension for various bounary conitions. Messaoui an Sai-Houari 5 prove the exponential stability in one-imensional nonlinear thermoelasticity with secon soun. Soufyane 6 establishe an exponential an polynomial ecay results of porous thermoelasticity incluing a memory term. Recently, Aouai an Soufyane 7 prove the polynomial an exponential stability for one-imensional problem in thermoelastic iffusion theory uner Fourier s law. To the author s knowlege, no work has been one regaring the exponential stability of the thermoelastic iffusion theory with secon soun though similar research in thermoelasticity has been popular in recent years. This paper will evote to stuy the exponential stability of the solution of the one-imensional thermoelastic iffusion theory uner Cattaneo s law. The moel that we consier is interesting not only because we take into account the thermaliffusion effect, but also because Cattaneo s law is physically more realistic than Fourier s law. In this case, the governing equations correspons to the coupling of three hyperbolic equations. This question is new in thermoelastic theories an poses new analytical an mathematical ifficulties. This kin of coupling has not been consiere previously, an we have a few results concerning the existence, uniqueness, an exponential ecay. For this reason,

3 International Journal of Differential Equations 3 the exponential ecay of the solution is very interesting an also very ifficult. We obtain the exponential ecay by the multiplier metho an constructing generalize Lyapunov functional. The remaining part of this paper is organize as follows: in Section, we give basic equations, an for completeness, we iscuss the well-poseness of the initial bounary value problem in a semigroup setting. In Section 3, we erive the various energy estimates, an we state the exponential ecay of the solution. In Section 4, we provie arguments for showing that the two systems, either τ >, τ > orτ τ, are close to each other, in the sense of energy estimates, of orer τ an τ.. Basic Equations an Preliminaries The governing equations for an isotropic, homogenous thermoelastic iffusion soli are as follows see 3 : i the equation of motion σ ij,j ρü i,. ii the stress-strain-temperature-iffusion relation σ ij μe ij δ ij λekk β θ β C ),. iii the isplacement-strain relation e ij ui,j u j,i ),.3 iv the energy equation q i,i ρt Ṡ,.4 v the Cattaneo s law for temperature kθ,i q i τ q i,.5 vi the entropy-strain-temperature-iffusion relation ρt S β T e kk ρc E θ at C,.6 vii the equation of conservation of mass η i,i Ċ,.7

4 4 International Journal of Differential Equations viii the Cattaneo s law for chemical potential ħp,i η i τ η i,.8 ix the chemical-strain-temperature-iffusion relation P β e kk aθ bc,.9 where β 3λ μ α t an β 3λ μ α c, α t,anα c are, respectively, the coefficients of linear thermal an iffusion expansion an λ an μ are Lamé s constants. θ T T is small temperature increment, T is the absolute temperature of the meium, an T is the reference uniform temperature of the boy chosen such that θ/t. q i is the heat conuction vector, k is the coefficient of thermal conuctivity, an c E is the specific heat at constant strain. σ ij are the components of the stress tensor, u i are the components of the isplacement vector, e ij are the components of the strain tensor, S is the entropy per unit mass, P is the chemical potential per unit mass, C is the concentration of the iffusive material in the elastic boy, ħ is the iffusion coefficient, η i enotes the flow of the iffusing mass vector, a is a measure of thermoiffusion effect, b is a measure of iffusive effect, an ρ is the mass ensity. τ is the thermal relaxation time, which will ensure that the heat conuction equation will preict finite spees of heat propagation. τ is the iffusion relaxation time, which will ensure that the equation satisfie by the concentration will also preict finite spees of propagation of matter from one meium to the other. We will now formulate a ifferent alternative form that will be useful in proving uniqueness in the next section. In this new formulation, we will use the chemical potential as a state variable instea of the concentration. From.9, weobtain C γ e kk np θ.. The alternative form can be written by substituting. into..8, σ ij,j ρü i, σ ij μe ij δ ij λ e kk γ θ γ P ), q i,i ρt Ṡ, kθ,i q i τ q i,. ρs cθ γ e kk P, η i,i Ċ, ħp,i η i τ η i,

5 International Journal of Differential Equations 5 where γ β aβ b, γ β b, λ λ β b, c ρc E T a b, a b, n b. are physical positive constants satisfying the following conition: cn >..3 Note that this conition implies that cθ θp np >..4 Conition.3 is neee to stabilize the thermoelastic iffusion system see 8 for more information on this. We assume throughout this paper that the conition.3 is satisfie. For the sake of simplicity, we assume that ρ, an we stuy the exponential stability in one-imension space. If u u x, t, θ θ x, t, anp P x, t escribe the isplacement, relative temperature an chemical potential, respectively, our equations take the form u tt αu xx γ θ x γ P x, in,l R, cθ t P t q x γ u xt, in,l R, τ q t q kθ x, in,l R,.5 θ t np t η x γ u xt, in,l R, τη t η ħη x, in,l R, where α λ μ >. The system is subjecte to the following initial conitions: u x, u x, u t x, u x, x,l, θ x, θ x, q x, q x, x,l,.6 P x, P x, η x, η x, x,l, an bounary conitions u,t u l, t, q,t q l, t, η,t η l, t, t..7

6 6 International Journal of Differential Equations For the sake of simplicity, we present a short irect iscussion of the the wellposeness for the linear initial bounary value.5.7. We transform the system.5.7 into a first-orer system of evolution type, finally applying semigroup theory. For a solution u, θ, q, P, η,letu be efine as αu x u t θ U, U, U q P η αu,x u θ q P η..8 The initial-bounary value problem.5.7 is equivalent to problem U t Q MU, U U,.9 where α x x γ x γ x c γ x x Q τ k, M.. x k n τ γ x x x ħ ħ We consier the Hilbert space U L,l 6 with inner prouct U, W U, QW L.. Let A : D A such that AU Q MU.. The omain of A is D A { U U,U,U 3,U 4,U 5,U 6) } T /U,U 4,U 6 H,l, U,U 3,U 5 H,l,.3

7 International Journal of Differential Equations 7 that is, U t AU, U U D A..4 On the other han, if U satisfies.4 for U efine in.8, then u,t u t U,s s, θ U 3, q U 4, P U 5, η U 6.5 satisfy.5.7 ; thatis,.4 an.5.7 are equivalent in the chosen spaces. The well-poseness is now a corollary of the following lemma characterizing A as a generator of a C -semigroup of contractions. Lemma.. i D A is ense in, an the operator A is issipative. ii A is close. iii D A D A, x x γ x γ x γ x x A W Q W..6 x k γ x x x ħ Proof. i The ensity of D A in is obvious, an we have Re AU, U k q x ħ η x..7 Then A is issipative. ii Let U n n D A, U n U, an AU n F, as n. Then, Φ : AU n, Φ F, Φ..8 Choosing successively Φ Φ,,,,, T, Φ H,l, Φ,, Φ 3,,, T, Φ 3 H,l, 3 Φ,,,, Φ 5, T, Φ 5 H,l, 4 Φ,,, Φ 4,, T, Φ 4 C,l,

8 8 International Journal of Differential Equations 5 Φ,,,,, Φ 6 T, Φ 6 C,l, 6 Φ, Φ,,, T, Φ C,l, we obtain U H,l an xu QF first component, U 4 H,l an γ x U x U 4 QF 3, 3 U 6 H,l an γ x U x U 6 QF 5, 4 U 3 H,l an x U 3 /ku 4 QF 4, 5 U 5 H,l an x U 5 /ħu 6 QF 6, 6 U H,l an x U γ x U 3 γ x U 5 QF. iii W D A F, Φ D A : AΦ, W Φ, F..9 Choosing Φ appropriately as in the proof of ii, the conclusion follows. With the Hille-Yosia theorem see 9 C -semigroups, we can state the following result. Theorem.. i The operator A is the infinitesimal generator of a C -semigroup of linear contractions T t e ta over the space for t. ii For any U D A, there exists a unique solution U t C, ; C, ; D A to.4 given by U t e ta U. iii If U D A n, n N, thenu t C, ; D A n an.4 yiels higher regularity in t. Moreover, we will use the Young inequality ±ab a δ δ 4 b, a, b R, δ >..3 The ifferential of.5 an.5 4 together with bounary conitions.7 yiels θ x, t x θ x x, P x, t x P x x, t..3 Then, θ an P efine by θ x, t : θ x, t l θ x x, P x, t : P x, t l P x x.3

9 International Journal of Differential Equations 9 satisfy with u, q, anη the same ifferential equations.5.7 as u, θ, q, P, η, but aitionally, we have the Poincaré inequality v x l v π xx,.33 for v θ,t as well as for v P, v u, v q or v η. In the sequel, we will work with θ an P but still write θ an P for simplicity until we will have prove Theorem 3.. From.5 3 an.5 5, we conclue θ xx τ k P xx τ ħ qt x q x, k ηt x η x. ħ.34 Finally, for the sake of simplicity, we will employ the same symbols C for ifferent constants, even in the same formula. In particular, we will enote by the same symbol C i ifferent constants ue to the use of Poincaré s inequality on the interval,l. 3. Exponential Stability Let u, θ, q, P, η be a solution to problem.5.7. Multiplying.5 by u t,.5 by θ,.5 3 by q,.5 4 by P,an.5 5 by η an integrating from to l,weget t E t q x η x, k ħ 3. where E t u t αu x cθ np θp τ k q τ ) ħ η x. 3. Differentiating.5 with respect to t, we get in the same manner t E t qt k x ηt ħ x, 3.3 where E t u tt αu xt cθ t np t θ t P t τ k q t τ ) ħ η t x. 3.4

10 International Journal of Differential Equations Let us efine the functionals F t G t α u xtu x 3 qu α tt 3 γ α qθ x 3γ qp x 3c θ x u t 3 ) P x u t x, αγ αγ αγ α u xtu x 3 ηu α tt 3 γ α ηp x 3γ ηθ x 3 P x u t 3n ) θ x u t x. αγ αγ αγ 3.5 Lemma 3.. Let u, θ, q, P, η be a solution to problem.5.7. Then, one has t F t 36α u tt ) απ l u t θ P x u 7 l xtx q α γ α t x 7c γ 3cγ 5γ ) αγ 6α l 3c l θ 36α π α xx γ 3cγ 5γ ) αγ 6α l 3c l P 36α π α xx, t G t 36α u tt ) απ l u t θ P x u 7 l xtx η α γ α t x 7n γ 3nγ 5γ ) αγ 6α l 3n l P 36α π α xx γ 3nγ 5γ ) αγ 6α l 3n l θ 36α π α xx. Proof. We will only prove 3.6 an 3.7 can be obtaine analogously. Multiplying.5 by u xx /α an using the Young inequality, we get u xxx α α t u ttx u x x γ θ x u xx x γ P x u xx x α α u xt u x x α u xt x 3γ θ 4α xx 3γ P 4α xx u 3 xxx, 3.8 which implies α t u xt u x x 3 u xxx α u xt x 3γ θ 4α xx 3γ P 4α xx. 3.9

11 International Journal of Differential Equations Multiplying.5 by 3u xt / αγ an using the Young inequality, we get 3 l u 3 l xtx qu xxt x 3c θ xt u t x 3 P xt u t x α αγ αγ αγ 3 l qu xx x 3 q t u xx x 3c l θ x u t x 3c θ x u tt x αγ t αγ αγ t αγ 3 l P x u t x 3 P x u tt x. αγ t αγ 3. Substituting.5 in the above equation, yiels 3 l u 3 l xtx α α γ t qu tt x 3 α t qθ x x 3γ l qp x x 3 q t u xx x αγ t αγ 3c l θ x u t x 3c θ x u xx x 3c θ αγ t γ α xx 3cγ θ x P x x αγ 3 l P x u t x 3 P x u xx x 3c θ x P x x 3cγ P αγ t γ α αγ xx. 3. Using the estimates 3 q t u xx x u αγ xxx 7 q γ α t x, 3c θ x u xx x u γ xxx 7c θxx, γ 3cγ αγ θ x P x x 3cγ θ αγ xx 3cγ P αγ xx, 3. 3 γ 3c α P x u xx x u xxx 7 Pxx, γ θ x P x x 3c θ α xx 3c P α xx,

12 International Journal of Differential Equations 3 l u xt α x t 3 qu α tt 3 γ α qθ x 3γ qp x 3c θ x u t 3 P x u t αγ αγ αγ ) 9cα l 7 q 3c γ α t x α γ γ γ θ xx 3n α ) x 4 9cα γ u xxx γ ) θ γ xx. 3.3 Combining 3.9 an 3.3, weget l F t u t xxx u 7 xtx α γ α q t x 3c 9cα α γ γ γ γ 4cα ) θ xx 3.4 3c 9 α α γ c γ γ γ 4cα ) P xx. Now, we conclue from.5 that ) u απ tt l u t θ P x ) α u xx γ θ x γ P x αγ u xx θ x αγ u xx P x γ γ θ x P x απ l u t θ P x ) ) 3α u xxx 3γ l θ π xx 3γ l P π xx α u xt x 3.5 whence u xxx 3α ) u tt α π u l t θ P x α γ ) l θ 3π xx ) γ l α P 3π xx u xt 3α x. 3.6 Combining 3.4 an 3.6, we get our conclusion follows.

13 International Journal of Differential Equations 3 Multiplying.5 by θ t an.5 4 by P t, an summing the results, yiels t ) l ) qθx ηp x x cθ t np t θ t P t x q t θ x x γ u xt θ t x η t P x x γ u xt P t x. 3.7 Using the estimates γ u xt θ t x c 4 θ t x γ c u xt x, t γ u xt P t x δ 4 P t x γ δ u xt x, θ t P t x c θ 4 t x Pt 4 c x, θt x qθx ηp x ) x c n δ 4 4 c ) Pt x 3.8 qt x ηt x θ xx γ Pxx c γ ) u xt δ x. Choosing δ such as n δ 4 4 c > n 3.9 yiels t ) c l qθx ηp x x θt x n Pt x qt x ηt x θ xx γ P xx c γ ) u xt δ x. 3. Now, we will show the main result of this section.

14 4 International Journal of Differential Equations Theorem 3.. Let u, θ, q, P, η beasolutiontoproblem.5.7. Then, the associate energy of first an secon orer E t E t E t j j t u t ) α j t u x ) c n j t P ) j t θ ) j t θ j t P τ k j t ) τ q ħ j t ) ) η x, t x 3. ecays exponentially; that is, c >, C >, t E t C E e c t. 3. Bouns for c an C can be given explicitly in terms of the coefficient α, γ,γ,c,n,,k, ħ,τ,τ, an l. Proof. Now, we efine the esire Lyapunov functional N t. For ε>, to be etermine later on, let N t ε E t F t G t ε qθx ηp x ) x. 3.3 Then, we conclue from an 3.4 that l N t q x η x t εk εħ 8α u tt ) απ l u t θ P x ) α εγ n εγ l u xt δ x εk 7 γ ε ) q α t x εħ 7 γ ε ) η α t x ε ) cθt np t x 3.4 ϖ ε ) θ xx ϱ ε ) P xx, where ϖ 7 c ) γ ϱ 7 n ) γ 3cγ 3nγ 5γ αγ αγ 3α l 8α π 3 c n, α 3cγ 3nγ 5γ αγ αγ 3α l 8α π 3 c n. α 3.5

15 International Journal of Differential Equations 5 Using.34,weobtain t N t k ε ϖ ε ) ) q x k ħ ε ϱ ε ) ) η x ħ ) 8α u tt απ l u t θ P x εk 7 γ ε α τ ϖ ε ) ) q k t x ) α εγ n εγ l u xt δ x 3.6 εħ 7 γ ε α ε τ ϱ ε ) ) η ħ t x ε ) cθt np t x. Using.3 an choosing <ε< such that all terms on the right-han sie of 3.7 become negative, α γ /c) γ /δ)) { k <ε<min ϖ, ħ ϱ, k / τ /k) ϖ 7/γ α)), } ħ / τ /ħ ϱ ) 7/γ α)). 3.7 Choosing ε as in 3.7, weobtainfrom t N t c u tt u t u xt θ P θ t P t q t η t q η ) x, 3.8 where c { min εk, εħ, 9α, } α,ε, 3.9 which implies t N t c E t, 3.3 with c c { min,α,c,,n, τ k, τ }. 3.3 ħ

16 6 International Journal of Differential Equations On the other han, we have ε >, C, C >, ε ε, t >:C E t N t C E t, 3.3 where C,C are etermine as follows. Let H t N t ε E t, 3.33 then H t C E t, 3.34 with { 3 c 3 n C max, αγ αγ 3 ), α γ γ 3ħ τ α γ α γ ) nγ, 6τ αγ αħ αk α, 3k τ α γ α γ )) cγ, αγ αk cγ ), 6τ ))} nγ. αħ 3.35 Choosing ε ε C, 3.36 we have C ε C, ε min{ε,ε } Moreover, from 3.3 an 3.3, we erive t N t c N t, 3.38 with c c C, 3.39 hence N t e c t N. 3.4

17 International Journal of Differential Equations 7 Applying 3.3 again, we have prove E t C E e c t, 3.4 with C C C, 3.4 an it hols. The copper material was chosen for purposes of numerical evaluations. The physical constants given by Table are foun in. Successively we can approximately compute ε,c,ε,ε,c,c,c,anc from the previous equations, getting finally c.68 56, 3.43 which inicates a slow ecay of the energy in the beginning but oes not mean that solutions o not ecay. Remark 3.3. In particular, we can get O τ,τ as τ,τ, Although the estimate for τ an τ are very coarse an might be far from being sharp, it inicates a slow ecay of the energy in usually measure time perios. The above relation of course oes not imply that solutions to the limiting case τ,τ, o not ecay. Instea, the ecay rate of the thermoiffusion system provies a better rate; that is, cθ t P t kθ xx, in,l R, θ t np t ħp xx, in,l R, 3.45 with initial conitions θ x, θ x, P x, P x, x,l, 3.46 an bounary conitions θ,t θ l, t, P,t P l, t, t In this case, we have l E t k θ x ħ P x, t 3.48

18 8 International Journal of Differential Equations Table : Values of the constants. μ 3.86 kg/ ms 3 λ 7.76 kg/ ms 3 ρ 8954 kg/m 3 c E 383.J/ Kg K α t.78 5 K k 386 W/ mk α c.98 4 m 3 /kg ħ.85 8 kg s/m 3 T 93 K a. 4 m / s K b.9 6 m 5 / s Kg τ s τ s l 6 4 m where E t cθ np θp τ k q τ ) ħ η x Using the Poincaré inequality, we get E t e νt E, 3.5 with ν π k l c ħ ) n The Limit Case τ,τ, We will show that the energy of the ifference of the solution u, θ, P, q, η to.5.7 an the the solution ũ, θ, P, q, η to the corresponing system with τ,τ, see 7 vanishes of orer τ τ as τ,τ,, provie the values at t coincie. For this purpose, let U, Θ,φ,Φ,ψ enote the ifference U u ũ, Θ θ θ, φ q q, Φ P P, ψ η η, 4. then U, Θ,φ,Φ,ψ satisfies U tt αu xx γ Θ x γ Φ x, cθ t Φ t φ x γ U xt, τ φ t φ kθ x τ k θ xt, 4. Θ t nφ t ψ x γ U xt, τψ t η ħψ x τħ P xt, U x,, U t x,, Θ x,, Φ x,, φ x,, ψ x,, x,l, 4.3

19 International Journal of Differential Equations 9 U,t U l, t, Θ,t Θ l, t, Φ,t Φ l, t. 4.4 Here, we assume the compatibility conitions q kθ,x, η ħp,x. 4.5 If E t enotes the energy of first orer for U, Θ,φ,Φ,ψ ; thatis, t E t φ x τ k θ xt φx ħ ψ x τ P xt ψx, 4.6 where E t Ut αu x cθ nφ ΘΦ τ k φ τ ) ħ ψ x. 4.7 Using the Young inequality, we obtain l E t φ x τ k θ xt t k x ψ x τ ħ l P xt ħ x. 4.8 Using initial conition 4.3 yiels E t τ k t θ xt x, s x s τ ħ t P xt x, s x s, 4.9 from where we get for T> fixe, t,t E t τ k T θ xt x, s x s τ ħ T P xt x, s x s. 4. Moreover, since θ xt x, s x s <, P xt x, s x s <, 4. because of the exponential ecay of the solution corresponing to the problem when τ,τ, see 7, we obtain a uniform boun on the right-han sie, C >, t :E t C τ τ ), 4.

20 International Journal of Differential Equations where C T T {k min θ xt x, s x s, ħ P xt }. x, s x s 4.3 Then we have E t O τ τ ) as τ,τ,, 4.4 also E t as t. 4.5 τ τ 5. Concluing Remarks By comparison of the approximate value of c with the value c.75 3 of the problem corresponing to τ,τ, compute in 7, we remark the secon value is significantly larger than the first. This confirms that thermoelastic moels with secon soun are physically more realistic than those given in the classic context. By comparison of the approximate value of c with the value ν.43 3 of the thermoiffusion problem, we conclue that the slow ecay of the elastic part is responsible for the low bouns on the ecay rates obtaine in this paper an in 7. 3 Finally, we remark that in 7, the exponential ecay of the solution was prove by means of the first energy only, while in our case, it is necessary to use secon-orer erivatives because of the more complicate system with secon soun. References H. W. Lor an Y. Shulman, A generalize ynamical theory of thermoelasticity, Journal of the Mechanics an Physics of Solis, vol. 5, no. 5, pp , 967. W. Nowacki, Dynamical problems of thermoelastic iffusion in solis I, Bulletin e l Acaémie Polonaise es Sciences Serie es Sciences Techniques, vol., p ; 9 35; 57 66, H. H. Sherief, F. A. Hamza, an H. A. Saleh, The theory of generalize thermoelastic iffusion, International Journal of Engineering Science, vol. 4, no. 5-6, pp , 4. 4 M. Aouai, Theory of generalize micropolar thermoelastic iffusion uner Lor-Shulman moel, Journal of Thermal Stresses, vol. 3, no. 9, pp , 9. 5 M. Aouai, The couple theory of micropolar thermoelastic iffusion, Acta Mechanica, vol. 8, no. 3-4, pp. 8 3, 9. 6 M. Aouai, A theory of thermoelastic iffusion materials with vois, Zeitschrift für Angewante Mathematik un Physik, vol. 6, no., pp ,. 7 M. Aouai, Qualitative results in the theory of thermoelastic iffusion mixtures, Journal of Thermal Stresses, vol. 33, no. 6, pp ,. 8 C. M. Dafermos, Contraction semigroups an trens to equilibrium in continuum mechanics, in Applications of Methos of Functional Analysis to Problems in Mechanics, P. Germain an B. Nayroles, Es., vol. 53 of Springer Lectures Notes in Mathematics, pp , Springer, Berlin, Germany, 976.

21 International Journal of Differential Equations 9 M. Slemro, Global existence, uniqueness, an asymptotic stability of classical smooth solutions in one-imensional nonlinear thermoelasticity, Archive for Rational Mechanics an Analysis, vol. 76, no., pp , 98. S. Jiang an R. Racke, Evolution Equations in Thermoelasticity, vol. of Monographs an Surveys in Pure an Applie Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA,. G. Lebeau an E. Zuazua, Decay rates for the three-imensional linear system of thermoelasticity, Archive for Rational Mechanics an Analysis, vol. 48, no. 3, pp. 79 3, 999. H. H. Sherief, On uniqueness an stability in generalize thermoelasticity, Quarterly of Applie Mathematics, vol. 44, no. 4, pp , M. A. Tarabek, On the existence of smooth solutions in one-imensional nonlinear thermoelasticity with secon soun, Quarterly of Applie Mathematics, vol. 5, no. 4, pp , R. Racke, Thermoelasticity with secon soun exponential stability in linear an non-linear -, Mathematical Methos in the Applie Sciences, vol. 5, no. 5, pp ,. 5 S. A. Messaoui an B. Sai-Houari, Exponential stability in one-imensional non-linear thermoelasticity with secon soun, Mathematical Methos in the Applie Sciences, vol. 8, no., pp. 5 3, 5. 6 A. Soufyane, Energy ecay for porous-thermo-elasticity systems of memory type, Applicable Analysis, vol. 87, no. 4, pp , 8. 7 M. Aouai an A. Soufyane, Polynomial an exponential stability for one-imensional problem in thermoelastic iffusion theory, Applicable Analysis, vol. 89, no. 6, pp ,. 8 M. Aouai, Generalize theory of thermoelastic iffusion for anisotropic meia, Journal of Thermal Stresses, vol. 3, no. 3, pp. 7 85, 8. 9 A. Pazy, Semigroups of Linear Operators an Applications to Partial Differential Equations, vol. 44 of Applie Mathematical Sciences, Springer, New York, NY, USA, 983. L. Thomas, Funamentals of Heat Transfer, Prentice-Hall, Englewoo Cliffs, NJ, USA, 98.

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