# Exponential Stability in Hyperbolic Thermoelastic Diffusion Problem with Second Sound

Authors: Moncef Aouadi
Journal: International Journal of Differential Equations
 (2011)

---

## Abstract

We consider a thermoelastic diffusion problem in one space dimension with second sound. The thermal and diffusion disturbances are modeled by Cattaneo's law for heat and diffusion equations to remove the physical paradox of infinite propagation speed 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 and mathematical difficulties. The exponential stability of the slightly damped and totally hyperbolic system is proved. Comparison with classical theory is given.


---

## Body

## 1. Introduction

The classical model for the propagation of heat turns into the well-known Fourier’s law(1.1)q+k∇θ=0,
where θ is temperature (difference to a fixed constant reference temperature), q is the heat conduction vector, and k is the coefficient of thermal conductivity. The model using classic Fourier's law inhibits the physical paradox of infinite propagation speed of signals. To eliminate this paradox, a generalized thermoelasticity theory has been developed subsequently. The development of this theory was accelerated by the advent of the second sound effects observed experimentally in materials at a very low temperature. In heat transfer problems involving very short time intervals and/or very high heat fluxes, it has been revealed that the inclusion of the second sound effects to the original theory yields results which are realistic and very much different from those obtained with classic Fourier’s law. The first theory was developed by Lord and Shulman [1]. In this theory, a modified law of heat conduction, the Cattaneo's law,
(1.2)τ0qt+q+k∇θ=0
replaces the classic Fourier's law. The heat equation associated with this a hyperbolic one and, hence, automatically eliminates the paradox of infinite speeds. The positive parameter τ0 is the relaxation time describing the time lag in the response of the heat flux to a gradient in the temperature.The development of high technologies in the years before, during, and after the second world war pronouncedly affected the investigations in which the fields of temperature and diffusion in solids cannot be neglected. The problems connected with the diffusion of matter in thermoelastic bodies and the interaction of mechanodiffusion processes have become the subject of research by many authors. At elevated and low temperatures, the processes of heat and mass transfer play a decisive role in many satellite problems, returning space vehicles, and landing on water or land. These days, oil companies are interested in the process of thermodiffusion for more efficient extraction of oil from oil deposits.Nowacki [2] developed the classic theory of thermoelastic diffusion under Fourier’s law. Sherief et al. [3] derived the theory of thermoelastic diffusion with second sound. Aouadi [4] derived the theory of micropolar thermoelastic diffusion under Cattaneo's law. Recently, Aouadi [5–7] derived the general equations of motion and constitutive equations of the linear thermoelastic diffusion theory in the context of different media, with uniqueness and existence theorems.In recent years, a relevant task has been developed to obtain exponential stability of solutions in thermoelastic theories. The classical theory was first considered by Dafermos [8] and Slemrod [9] and it has been studied in the book of Jiang and Racke [10] and the contribution of Lebeau and Zuazua [11]. One should mention a paper by Sherief [12], where the stability of the null solution also in higher dimension is proved. We mention also the work of Tarabek [13] who studied even one dimensional nonlinear systems and obtained the strong convergence of derivatives of solutions to zero. Racke [14] proved the exponential decay of linear and nonlinear thermoelastic systems with second sound in one dimension for various boundary conditions. Messaoudi and Said-Houari [15] proved the exponential stability in one-dimensional nonlinear thermoelasticity with second sound. Soufyane [16] established an exponential and polynomial decay results of porous thermoelasticity including a memory term.Recently, Aouadi and Soufyane [17] proved the polynomial and exponential stability for one-dimensional problem in thermoelastic diffusion theory under Fourier's law. To the author's knowledge, no work has been done regarding the exponential stability of the thermoelastic diffusion theory with second sound though similar research in thermoelasticity has been popular in recent years. This paper will devote to study the exponential stability of the solution of the one-dimensional thermoelastic diffusion theory under Cattaneo's law. The model that we consider is interesting not only because we take into account the thermal-diffusion effect, but also because Cattaneo's law is physically more realistic than Fourier’s law. In this case, the governing equations corresponds to the coupling of three hyperbolic equations. This question is new in thermoelastic theories and poses new analytical and mathematical difficulties. This kind of coupling has not been considered previously, and we have a few results concerning the existence, uniqueness, and exponential decay. For this reason, the exponential decay of the solution is very interesting and also very difficult. We obtain the exponential decay by the multiplier method and constructing generalized Lyapunov functional.The remaining part of this paper is organized as follows: in Section2, we give basic equations, and for completeness, we discuss the well-posedness of the initial boundary value problem in a semigroup setting. In Section 3, we derive the various energy estimates, and we state the exponential decay of the solution. In Section 4, we provide arguments for showing that the two systems, either τ0>0,τ>0 or τ0=τ=0, are close to each other, in the sense of energy estimates, of order τ02 and τ2.


## 2. Basic Equations and Preliminaries

The governing equations for an isotropic, homogenous thermoelastic diffusion solid are as follows (see [3]):(i)
the equation of motion(2.1)σij,j=ρüi,(ii)
the stress-strain-temperature-diffusion relation(2.2)σij=2μeij+δij(λekk-β1θ-β2C),(iii)
the displacement-strain relation(2.3)eij=12(ui,j+uj,i),(iv)
the energy equation(2.4)qi,i=-ρT0Ṡ,(v)
the Cattaneo's law for temperature(2.5)-kθ,i=qi+τ0q̇i,(vi)
the entropy-strain-temperature-diffusion relation(2.6)ρT0S=β1T0ekk+ρcEθ+aT0C,(vii)
the equation of conservation of mass(2.7)ηi,i=-Ċ,(viii)
the Cattaneo's law for chemical potential(2.8)-ℏP,i=ηi+τη̇i,(ix)
the chemical-strain-temperature-diffusion relation(2.9)P=-β2ekk-aθ+bC,
where β1=(3λ+2μ)αt and β2=(3λ+2μ)αc, αt, and αc are, respectively, the coefficients of linear thermal and diffusion expansion and λ and μ are Lamé’s constants. θ=T-T0 is small temperature increment, T is the absolute temperature of the medium, and  T0 is the reference uniform temperature of the body chosen such that |θ/T0|≪1. qi is the heat conduction vector, k is the coefficient of thermal conductivity, and cE is the specific heat at constant strain. σij are the components of the stress tensor, ui are the components of the displacement vector, eij 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 diffusive material in the elastic body, ℏ is the diffusion coefficient, ηi denotes the flow of the diffusing mass vector, “a” is a measure of thermodiffusion effect, “b” is a measure of diffusive effect, and ρ is the mass density. τ0 is the thermal relaxation time, which will ensure that the heat conduction equation will predict finite speeds of heat propagation. τ is the diffusion relaxation time, which will ensure that the equation satisfied by the concentration will also predict finite speeds of propagation of matter from one medium to the other.We will now formulate a different 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 instead of the concentration. From (2.9), we obtain
(2.10)C=γ2ekk+nP+dθ.
The alternative form can be written by substituting (2.10) into (2.1)–(2.8),
(2.11)σij,j=ρüi,σij=2μeij+δij(λ0ekk-γ1θ-γ2P),qi,i=-ρT0Ṡ,-kθ,i=qi+τ0q̇i,ρS=cθ+γ1ekk+dP,ηi,i=-Ċ,-ℏP,i=ηi+τη̇i,
where
(2.12)γ1=β1+aβ2b,γ2=β2b,λ0=λ-β22b,c=ρcET0+a2b,d=ab,n=1b
are physical positive constants satisfying the following condition:
(2.13)cn-d2>0.
Note that this condition implies that
(2.14)cθ2+2dθP+nP2>0.
Condition (2.13) is needed to stabilize the thermoelastic diffusion system (see [18] for more information on this).We assume throughout this paper that the condition (2.13) is satisfied.For the sake of simplicity, we assume thatρ=1, and we study the exponential stability in one-dimension space. If u=u(x,t), θ=θ(x,t), and P=P(x,t) describe the displacement, relative temperature and chemical potential, respectively, our equations take the form
(2.15)utt-αuxx+γ1θx+γ2Px=0,in]0,l[×R+,cθt+dPt+qx+γ1uxt=0,in]0,l[×R+,τ0qt+q+kθx=0,in]0,l[×R+,dθt+nPt+ηx+γ2uxt=0,in]0,l[×R+,τηt+η+ℏηx=0,in]0,l[×R+,
where α=λ0+2μ>0. The system is subjected to the following initial conditions:
(2.16)u(x,0)=u0(x),ut(x,0)=u1(x),x∈]0,l[,θ(x,0)=θ0(x),q(x,0)=q0(x),x∈]0,l[,P(x,0)=P0(x),η(x,0)=η0(x),x∈]0,l[,
and boundary conditions
(2.17)u(0,t)=u(l,t)=0,q(0,t)=q(l,t)=0,η(0,t)=η(l,t)=0,t≥0.For the sake of simplicity, we present a short direct discussion of the the well-posedness for the linear initial boundary value (2.15)1–(2.17). We transform the system (2.15)1–(2.17) into a first-order system of evolution type, finally applying semigroup theory. For a solution (u,θ,q,P,η), let 𝒰 be defined as
(2.18)U=(αuxutθqPη),U(0,⋅)=U0=(αu0,xu1θ0q0P0η0).The initial-boundary value problem (2.15)1–(2.17) is equivalent to problem
(2.19)dUdt+Q-1MU=0,U(0)=U0,
where
(2.20)Q=(1α0000001000000c0d0000τ0k0000d0n000000τℏ),M=(0-∂x0000-∂x0γ1∂x0γ2∂x00γ1∂x0∂x0000∂x1k000γ2∂x000∂x0000∂x1ℏ).We consider the Hilbert space£=𝒰∈(L2(0,ℓ))6 with inner product
(2.21)〈U,W〉£=〈U,QW〉L2.
Let 𝒜:𝒟(𝒜)⊂£→£ such that
(2.22)AU=Q-1MU.
The domain of 𝒜 is
(2.23)D(A)={U=(U1,U2,U3,U4,U5,U6)T∈£/U2,U4,U6∈H01(0,l),U1,U3,U5∈H1(0,l)},
that is,
(2.24)dUdt+AU=0,U(0)=U0∈D(A).On the other hand, if𝒰 satisfies (2.24) for 𝒰0 defined in (2.18), then
(2.25)u(⋅,t)=u0(⋅)+∫0tU2(⋅,s)ds,θ=U3,q=U4,P=U5,η=U6
satisfy (2.15)1–(2.17); that is, (2.24) and (2.15)1–(2.17) are equivalent (in the chosen spaces). The well-posedness is now a corollary of the following lemma characterizing 𝒜 as a generator of a C0-semigroup of contractions.Lemma 2.1.
(i)𝒟(𝒜) is dense in £, and the operator -𝒜 is dissipative.
(ii)𝒜 is closed.
(iii)(2.26)D(A*)=D(A),A*W=Q-1(0∂x0000∂x0-γ1∂x0-γ2∂x00-γ1∂x0-∂x0000-∂x1k000-γ2∂x000-∂x0000-∂x1ℏ)W.Proof.
(i) The density of𝒟(𝒜) in £ is obvious, and we have
(2.27)Re〈-AU,U〉£=-1k∫0lq2dx-1ℏ∫0lη2dx≤0.
Then -𝒜 is dissipative.
(ii) Let(𝒰n)n⊂𝒟(𝒜),𝒰n→𝒰∈£, and 𝒜𝒰n→ℱ∈£, as n→∞. Then,
(2.28)∀Φ∈£:〈AUn,Φ〉£⟶〈F,Φ〉£.
Choosing successively(1)
Φ=(Φ1,0,0,0,0,0)T,Φ1∈H1(0,ℓ),(2)
Φ=(0,0,Φ3,0,0,0)T,Φ3∈H1(0,ℓ),(3)
Φ=(0,0,0,0,Φ5,0)T,Φ5∈H1(0,ℓ),(4)
Φ=(0,0,0,Φ4,0,0)T,Φ4∈C0∞(0,ℓ),(5)
Φ=(0,0,0,0,0,Φ6)T,Φ6∈C0∞(0,ℓ),(6)
Φ=(0,Φ2,0,0,0)T,Φ2∈C0∞(0,ℓ),we obtain(1)
U2∈H01(0,ℓ) and -∂xU2=[𝒬ℱ]1 (first component),(2)
U4∈H01(0,ℓ) and γ1∂xU2+∂xU4=[𝒬ℱ]3,(3)
U6∈H01(0,ℓ) and γ2∂xU2+∂xU6=[𝒬ℱ]5,(4)
U3∈H1(0,ℓ) and ∂xU3+1/kU4=[𝒬ℱ]4,(5)
U5∈H1(0,ℓ) and ∂xU5+1/ℏU6=[𝒬ℱ]6,(6)
U1∈H1(0,ℓ) and -∂xU1+γ1∂xU3+γ1∂xU5=[𝒬ℱ]2.
(iii)(2.29)W∈D(A*)⟺∃F∈£,∀Φ∈D(A):〈AΦ,W〉£⟶〈Φ,F〉£.
Choosing Φ appropriately as in the proof of (ii), the conclusion follows.With the Hille-Yosida theorem (see [19]) C0-semigroups, we can state the following result.Theorem 2.2.
(i)  The operator-𝒜 is the infinitesimal generator of a C0-semigroup of linear contractions T(t)=e-t𝒜 over the space £ for t≥0.
(ii) For any𝒰0∈𝒟(𝒜), there exists a unique solution 𝒰(t)∈C1([0,∞);£)∩C0([0,∞);𝒟(𝒜))to (2.24) given by 𝒰(t)=e-t𝒜𝒰0.
(iii) If𝒰0∈𝒟(𝒜n), n∈ℕ, then 𝒰(t)∈C0([0,∞);𝒟(𝒜n)) and (2.24) yields higher regularity in t.Moreover, we will use the Young inequality(2.30)±ab≤a2δ+δ4b2,∀a,b∈R,δ>0.The differential of (2.15)2 and (2.15)4 together with boundary conditions (2.17) yields
(2.31)∫0lθ(x,t)dx=∫0lθ0(x)dx,∫0lP(x,t)dx=∫0lP0(x)dx,t≥0.
Then,  θ¯ and P¯ defined by
(2.32)θ¯(x,t):=θ(x,t)-1l∫0lθ0(x)dx,P¯(x,t):=P(x,t)-1l∫0lP0(x)dx
satisfy with u,q, and η the same differential equations (2.15)1–(2.17) as (u,θ,q,P,η), but additionally, we have the Poincaré inequality
(2.33)∫0lv2dx≤l2π2∫0lvx2dx,
for v=θ¯(·,t) as well as for v=P¯, v=u, v=q or v=η.In the sequel, we will work withθ¯ and P¯ but still write θ and P for simplicity until we will have proved Theorem 3.2.From (2.15)3 and (2.15)5, we conclude
(2.34)∫0lθx2dx≤2τ02k2∫0lqt2dx+2k2∫0lq2dx,∫0lPx2dx≤2τ2ℏ2∫0lηt2dx+2ℏ2∫0lη2dx.
Finally, for the sake of simplicity, we will employ the same symbols C for different constants, even in the same formula. In particular, we will denote by the same symbol Ci different constants due to the use of Poincaré's inequality on the interval [0,ℓ].


## 3. Exponential Stability

Let(u,θ,q,P,η) be a solution to problem (2.15)1–(2.17). Multiplying (2.15)1 by ut, (2.15)2 by θ, (2.15)3 by q, (2.15)4 by P, and (2.15)5 by η and integrating from 0 to ℓ, we get
(3.1)ddtE1(t)=-1k∫0lq2dx-1ℏ∫0lη2dx,
where
(3.2)E1(t)=12∫0l(ut2+αux2+cθ2+nP2+2dθP+τ0kq2+τℏη2)dx.
Differentiating (2.15) with respect to t, we get in the same manner
(3.3)ddtE2(t)=-1k∫0lqt2dx-1ℏ∫0lηt2dx,
where
(3.4)E2(t)=12∫0l(utt2+αuxt2+cθt2+nPt2+2dθtPt+τ0kqt2+τℏηt2)dx.Let us define the functionals(3.5)F(t)=-∫0l(1αuxtux+3α2γ1qutt+3αqθx+3γ2αγ1qPx+3cαγ1θxut+3dαγ1Pxut)dx,G(t)=-∫0l(1αuxtux+3α2γ2ηutt+3αηPx+3γ1αγ2ηθx+3dαγ2Pxut+3nαγ2θxut)dx.Lemma 3.1.
Let(u,θ,q,P,η) be a solution to problem (2.15)1–(2.17). Then, one has
(3.6)ddtF(t)≤-136α2∫0l(utt2+απ2l2ut2+θ2+P2)dx-1α∫0luxt2dx+27γ12α2∫0lqt2dx+(27c2γ12+3cγ22αγ1+5γ126α2+l236α2π2+3c2α)∫0lθx2dx+(27d2γ22+3cγ22αγ1+5γ226α2+l236α2π2+3c2α)∫0lPx2dx,(3.7)ddtG(t)≤-136α2∫0l(utt2+απ2l2ut2+θ2+P2)dx-1α∫0luxt2dx+27γ22α2∫0lηt2dx+(27n2γ22+3nγ12αγ2+5γ226α2+l236α2π2+3n2α)∫0lPx2dx+(27d2γ12+3nγ12αγ2+5γ126α2+l236α2π2+3n2α)∫0lθx2dx.Proof.
We will only prove (3.6) and (3.7) can be obtained analogously. Multiplying (2.15)1 by uxx/α and using the Young inequality, we get
(3.8)∫0luxx2dx=-1α∫0luttxuxdx+γ1α∫0lθxuxxdx+γ2α∫0lPxuxxdx≤-1αddt∫0luxtuxdx+1α∫0luxt2dx+3γ124α2∫0lθx2dx+3γ224α2∫0lPx2dx+23∫0luxx2dx,
which implies
(3.9)1αddt∫0luxtuxdx≤-13∫0luxx2dx+1α∫0luxt2dx+3γ124α2∫0lθx2dx+3γ224α2∫0lPx2dx.
Multiplying (2.15)2 by 3uxt/(αγ1) and using the Young inequality, we get
(3.10)3α∫0luxt2dx=3αγ1∫0lquxxtdx+3cαγ1∫0lθxtutdx+3dαγ1∫0lPxtutdx=3αγ1ddt∫0lquxxdx-3αγ1∫0lqtuxxdx+3cαγ1ddt∫0lθxutdx-3cαγ1∫0lθxuttdx+3dαγ1ddt∫0lPxutdx-3dαγ1∫0lPxuttdx.
Substituting (2.15)1 in the above equation, yields
(3.11)3α∫0luxt2dx=3α2γ1ddt∫0lquttdx+3αddt∫0lqθxdx+3γ2αγ1ddt∫0lqPxdx-3αγ1∫0lqtuxxdx+3cαγ1ddt∫0lθxutdx-3cγ1∫0lθxuxxdx+3cα∫0lθx2dx+3cγ2αγ1∫0lθxPxdx+3dαγ1ddt∫0lPxutdx-3dγ1∫0lPxuxxdx+3cα∫0lθxPxdx+3cγ2αγ1∫0lPx2dx.
Using the estimates(3.12)-3αγ1∫0lqtuxxdx≤112∫0luxx2dx+27γ12α2∫0lqt2dx,-3cγ1∫0lθxuxxdx≤112∫0luxx2dx+27c2γ12∫0lθx2dx,3cγ2αγ1∫0lθxPxdx≤3cγ22αγ1∫0lθx2dx+3cγ22αγ1∫0lPx2dx,-3dγ2∫0lPxuxxdx≤112∫0luxx2dx+27d2γ22∫0lPx2dx,3cα∫0lθxPxdx≤3c2α∫0lθx2dx+3c2α∫0lPx2dx,(3.13)3α∫0luxt2dx≤ddt∫0l(3α2γ1qutt+3αqθx+3γ2αγ1qPx+3cαγ1θxut+3dαγ1Pxut)dx+14∫0luxx2dx+27γ12α2∫0lqt2dx+3cα(9cαγ12+γ22γ1+12)∫0lθx2dx+3nα(9cαγ22+γ12γ2+12)∫0lθx2dx.
Combining (3.9) and (3.13), we get
(3.14)ddtF(t)≤-112∫0luxx2dx-2α∫0luxt2dx+27γ12α2∫0lqt2dx+3cα(9cαγ12+γ22γ1+γ124cα+12)∫0lθx2dx+3cα(9d2αγ22c+γ22γ1+γ224cα+12)∫0lPx2dx.
Now, we conclude from (2.15) that
(3.15)∫0l(utt2+απ2l2ut2+θ2+P2)dx=∫0l(α2uxx2+γ12θx2+γ22Px2-2αγ1uxxθx-2αγ2uxxPx+2γ1γ2θxPx+απ2l2ut2+θ2+P2)dx≤3α2∫0luxx2dx+(3γ12+l2π2)∫0lθx2dx+(3γ22+l2π2)∫0lPx2dx+α∫0luxt2dx
whence
(3.16)-∫0luxx2dx≤-13α2∫0l(utt2+α2π2l2ut2+θ2+P2)dx+1α2(γ12+l23π2)∫0lθx2dx+1α2(γ22+l23π2)∫0lPx2dx+13α∫0luxt2dx.
Combining (3.14) and (3.16), we get our conclusion follows.Multiplying(2.15)2 by θt and (2.15)4 by Pt, and summing the results, yields
(3.17)-ddt∫0l(qθx+ηPx)dx=-∫0l(cθt2+nPt2+2dθtPt)dx-∫0lqtθxdx-γ1∫0luxtθtdx-∫0lηtPxdx-γ2∫0luxtPtdx.Using the estimates(3.18)γ1∫0luxtθtdx≤c4∫0lθt2dx+γ12c∫0luxt2dx,γ2∫0luxtPtdx≤δ4∫0lPt2dx+γ22δ∫0luxt2dx,-2d∫0lθtPtdx≤c4∫0lθt2dx+4d2c∫0lPt2dx,-ddt∫0l(qθx+ηPx)dx≤-c2∫0lθt2dx-(n-δ4-4d2c)∫0lPt2dx+12∫0lqt2dx+12∫0lηt2dx+12∫0lθx2dx+12∫0lPx2dx+(γ12c+γ22δ)∫0luxt2dx.Choosingδ such as
(3.19)n-δ4-4d2c>n2
yields
(3.20)-ddt∫0l(qθx+ηPx)dx≤-c2∫0lθt2dx-n2∫0lPt2dx+12∫0lqt2dx+12∫0lηt2dx+12∫0lθx2dx+12∫0lPx2dx+(γ12c+γ22δ)∫0luxt2dx.
Now, we will show the main result of this section.Theorem 3.2.
Let(u,θ,q,P,η) be a solution to problem (2.15)1–(2.17). Then, the associated energy of first and second order
(3.21)E(t)=E1(t)+E2(t)=12∑j=12∫0l((∂tj-1ut)2+α(∂tj-1ux)2+c(∂tj-1θ)2+n(∂tj-1P)2+2d∂tj-1θ∂tj-1P+τ0k(∂tj-1q)2+τℏ(∂tj-1η)2)(x,t)dx
decays exponentially; that is,
(3.22)∃c0>0,∃C0>0,∀t≥0E(t)≤C0E(0)e-c0t.
Bounds forc0 and C0 can be given explicitly in terms of the coefficient α,γ1,γ2,c,n,d,k,ℏ,τ0,τ, and ℓ.Proof.
Now, we define the desired Lyapunov functional𝒩(t). For ɛ>0, to be determined later on, let
(3.23)N(t)=1ɛE(t)+F(t)+G(t)-ɛ∫0l(qθx+ηPx)dx.
Then, we conclude from (3.1)–(3.6) and (3.14) that
(3.24)ddtN(t)≤-1ɛk∫0lq2dx-1ɛℏ∫0lη2dx-118α2∫0l(utt2+απ2l2ut2+θ2+P2)dx-(2α-ɛγ22n-ɛγ12δ)∫0luxt2dx-(1ɛk-27γ12α2-ɛ2)∫0lqt2dx-(1ɛℏ-27γ22α2-ɛ2)∫0lηt2dx-ɛ2∫0l(cθt2+nPt2)dx+(ϖ+ɛ2)∫0lθx2dx+(ϱ+ɛ2)∫0lPx2dx,
where
(3.25)ϖ=27(c2+d2)γ12+3cγ22αγ1+3nγ12αγ2+5γ123α2+l218α2π2+32α(c+n),ϱ=27(n2+d2)γ12+3cγ22αγ1+3nγ12αγ2+5γ223α2+l218α2π2+32α(c+n).
Using (2.34), we obtain
(3.26)ddtN(t)≤-1k(1ɛ-2k(ϖ+ɛ2))∫0lq2dx-1ℏ(1ɛ-2ℏ(ϱ+ɛ2))∫0lη2dx-118α2∫0l(utt2+απ2l2ut2+θ2+P2)dx-(2α-ɛγ22n-ɛγ12δ)∫0luxt2dx-(1ɛk-27γ12α2-ɛ2-2τ02k2(ϖ+ɛ2))∫0lqt2dx-(1ɛℏ-27γ22α2-ɛ2-ɛ2-2τ02ℏ2(ϱ+ɛ2))∫0lηt2dx-ɛ2∫0l(cθt2+nPt2)dx.
Using (2.13) and choosing 0<ɛ<1 such that all terms on the right-hand side of (3.27) become negative,
(3.27)2α((γ12/c)+(γ22/δ))<ɛ<min⁡⁡{k1+2ϖ,ℏ1+2ϱ,1k((1/2)+(τ02/k2)(1+2ϖ)+(27/γ12α2)),1ℏ((1/2)+(τ2/ℏ2)(1+2ϱ)+(27/γ22α2))}.
Choosing ɛ as in (3.27), we obtain from
(3.28)ddtN(t)≤-c1∫0l(utt2+ut2+uxt2+θ2+P2+θt2+Pt2+qt2+ηt2+q2+η2)dx,
where
(3.29)c1=12min⁡{1ɛk,1ɛℏ,19α2,2α,ɛ},
which implies
(3.30)ddtN(t)≤-c2E(t),
with
(3.31)c2=c12min⁡⁡{1,α,c,d,n,τ0k,τℏ}.
On the other hand, we have(3.32)∃ɛ2>0,∃C1,C2>0,∀ɛ≤ɛ2,∀t>0:C1E(t)≤N(t)≤C2E(t),
where C1,C2 are determined as follows. Let
(3.33)H(t)=N(t)-1ɛE(t),
then
(3.34)|H(t)|≤C1E(t),
with
(3.35)C1=max⁡⁡{3(c+d)αγ1+3(n+d)αγ2,3α2(1γ1+1γ2),1α2,3kτ0(1α2γ1+1α+γ2αγ1+2αk2(1+cγ1)),3ℏτ(1α2γ2+1α+γ1αγ2+2αℏ2(1+nγ2),6τ0αk(1+cγ1),6ταℏ(1+nγ2))}.
Choosing(3.36)ɛ≤ɛ2=12C1,
we have
(3.37)C2=1ɛ+C1,ɛ=min⁡⁡{ɛ1,ɛ2}.
Moreover, from (3.30) and (3.32), we derive
(3.38)ddtN(t)≤-c0N(t),
with
(3.39)c0=c2C2,
hence
(3.40)N(t)≤e-c0tN(0).
Applying (3.32) again, we have proved
(3.41)E(t)≤C0E(0)e-c0t,
with
(3.42)C0=C2C1,
and it holds.The copper material was chosen for purposes of numerical evaluations. The physical constants given by Table1 are found in [20].Table 1
Values of the constants.
μ=3.86×1010kg/(ms3)λ=7.76×1010kg/(ms3)ρ=8954kg/m3cE=383.1J/(KgK)αt=1.78×10-5K-1k=386W/(mK)αc=1.98×10-4m3/kgℏ=0.85×10-8kgs/m3T0=293Ka=1.2×104m2/(s2K)b=0.9×106m5/(s2Kg)τ0=10-12sτ=10-11sℓ=6×10-4mSuccessively we can approximately computeɛ1,C1,ɛ2,ɛ,c1,c2,C0, and c0 from the previous equations, getting finally
(3.43)c0≈2.68×10-56,
which indicates a slow decay of the energy in the beginning but does not mean that solutions do not decay.Remark 3.3.
In particular, we can get(3.44)d0=O(τ0,τ)as(τ0,τ)⟶(0,0).
Although the estimate for τ0 and τ are very coarse and might be far from being sharp, it indicates a slow decay of the energy in usually measured time periods. The above relation of course does not imply that solutions to the limiting case (τ0,τ)=(0,0) do not decay. Instead, the decay rate of the thermodiffusion system provides a better rate; that is,
(3.45)cθt+dPt-kθxx=0,in]0,l[×R+,dθt+nPt-ℏPxx=0,in]0,l[×R+,
with initial conditions
(3.46)θ(x,0)=θ0(x),P(x,0)=P0(x),x∈]0,l[,
and boundary conditions
(3.47)θ(0,t)=θ(l,t)=0,P(0,t)=P(l,t)=0,t≥0.
In this case, we have
(3.48)ddtE(t)=-k∫0lθ2dx-ℏ∫0lP2dx,
where
(3.49)E(t)=12∫0l(cθ2+nP2+2dθP+τ0kq2+τℏη2)dx.
Using the Poincaré inequality, we get
(3.50)E(t)≤e-νtE(0),
with
(3.51)ν=2π2l2(kc+ℏn)≈2.43×10-3.


## 4. The Limit Case(τ0,τ)→(0,0)

We will show that the energy of the difference of the solution(u,θ,P,q,η) to (2.15)1–(2.17) and the the solution (ũ,θ̃,P̃,q̃,η̃) to the corresponding system with (τ0,τ)=(0,0) (see [17]) vanishes of order τ02+τ2 as (τ0,τ)→(0,0), provided the values at t=0 coincide. For this purpose, let (U,Θ,ϕ,Φ,ψ) denote the difference
(4.1)U=u-ũ,Θ=θ-θ̃,ϕ=q-q̃,Φ=P-P̃,ψ=η-η̃,
then (U,Θ,ϕ,Φ,ψ) satisfies
(4.2)Utt-αUxx+γ1Θx+γ2Φx=0,cΘt+dΦt+ϕx+γ1Uxt=0,τ0ϕt+ϕ+kΘx=τ0kθ̃xt,dΘt+nΦt+ψx+γ2Uxt=0,τψt+η+ℏψx=τℏP̃xt,(4.3)U(x,0)=0,Ut(x,0)=0,Θ(x,0)=0,Φ(x,0)=0,ϕ(x,0)=0,ψ(x,0)=0,x∈]0,l[,(4.4)U(0,t)=U(l,t)=0,Θ(0,t)=Θ(l,t)=0,Φ(0,t)=Φ(l,t)=0.
Here, we assumed the compatibility conditions
(4.5)q0=-kθ0,x,η0=-ℏP0,x.
If E(t) denotes the energy of first order for (U,Θ,ϕ,Φ,ψ); that is,
(4.6)ddtE(t)=-1k∫0lϕ2dx+τ0∫0lθ̃xtϕdx-1ℏ∫0lψ2dx+τ∫0lP̃xtψdx,
where
(4.7)E(t)=12∫0l(Ut2+αUx2+cΘ2+nΦ2+2dΘΦ+τ0kϕ2+τℏψ2)dx.
Using the Young inequality, we obtain
(4.8)ddtE(t)≤-12k∫0lϕ2dx+τ02k2∫0lθ̃xt2dx-12ℏ∫0lψ2dx+τ2ℏ2∫0lP̃xt2dx.Using initial condition (4.3) yields
(4.9)E(t)≤τ02k2∫0t∫0lθ̃xt2(x,s)dxds+τ2ℏ2∫0t∫0lP̃xt2(x,s)dxds,
from where we get for T>0 fixed, t∈[0,T](4.10)E(t)≤τ02k2∫0T∫0lθ̃xt2(x,s)dxds+τ2ℏ2∫0T∫0lP̃xt2(x,s)dxds.
Moreover, since
(4.11)∫0∞∫0lθ̃xt2(x,s)dxds<∞,∫0∞∫0lP̃xt2(x,s)dxds<∞,
because of the exponential decay of the solution corresponding to the problem when (τ0,τ)=(0,0) (see [17]), we obtain a uniform bound on the right-hand side,
(4.12)∃C>0,∀t≥0:E(t)≤C(τ02+τ2),
where
(4.13)C=12min⁡⁡{k∫0T∫0lθ̃xt2(x,s)dxds,ℏ∫0T∫0lP̃xt2(x,s)dxds}.
Then we have
(4.14)E(t)≤O(τ02+τ2)as(τ0,τ)⟶(0,0),
also
(4.15)E(t)τ02+τ2⟶0ast⟶0.


## 5. Concluding Remarks

(1)
By comparison of the approximate value ofc0≈2.68×10-56 with the value c0≈1.75×10-13 of the problem corresponding to (τ0,τ)=(0,0) computed in [17], we remark the second value is significantly larger than the first. This confirms that thermoelastic models with second sound are physically more realistic than those given in the classic context.(2)
By comparison of the approximate value ofc0≈2.68×10-56 with the value ν≈2.43×10-3 of the thermodiffusion problem, we conclude that the slow decay of the elastic part is responsible for the low bounds on the decay rates obtained in this paper and in [17].(3)
Finally, we remark that in [17], the exponential decay of the solution was proved by means of the first energy only, while in our case, it is necessary to use second-order derivatives because of the more complicated system with second sound.



---

*Source: 274843-2011-09-18.xml*