Exponential Stability of Swelling Soils with Thermodiffusion Effects

Abstract

In this work, we study a one-dimensional coupled hyperbolic–parabolic system modeling the dynamics of swelling soils under thermodiffusion effects. The model describes the interaction between the deformation of the solid skeleton, the pore fluid motion, the temperature variation, and a diffusive process formulated through chemical potential. Under mixed boundary conditions and without introducing additional mechanical damping or imposing restrictive relations among the physical parameters, we prove exponential stability of the system. Our analysis is based on the energy method. In contrast to the standard energy functional commonly used in related thermodiffusion models, we introduce a modified positive energy functional better adapted to the coupled structure of the system. By combining this energy with suitable auxiliary functionals, we construct an appropriate Lyapunov functional and derive an exponential stability estimate. Our result shows that thermodiffusion alone yields sufficient dissipation for exponential stabilization, complementing earlier works where exponential stability requires extra damping mechanisms or equal wave-speed assumptions.

1. Introduction

Swelling (expansive) soils are among the most problematic geomaterials in geotechnical engineering, chiefly because they undergo pronounced volume changes under fluctuations in moisture content and temperature. As reported in the classical engineering literature, this cyclic swelling–shrinkage mechanism typically produces surface heaving in wet periods and contraction-induced fissuring during drying, thereby accelerating cracking, differential settlement, and progressive deterioration of structural components such as pavements, shallow foundations, buried pipelines, and building slabs; see, for instance, Handy [1]. The severity of these effects becomes particularly apparent in climates with strong seasonal alteration between wet and dry periods. Comprehensive discussions of expansive soil behavior and its engineering consequences can be found in the classical monograph of Chen [2], as well as in the applied engineering references by Leonard [3] and by Jones and Jefferson [4].

From the standpoint of mathematical modeling, swelling soils are commonly described within the theory of poroelasticity, where the deformation of the solid skeleton is coupled with the motion of pore fluids. The conceptual and analytical foundations of this framework can be traced back to the seminal contributions of Biot [5,6] on consolidation and wave propagation in fluid-saturated porous media. Related formulations also arise in the continuum theory of elastic solids with voids, as developed by Goodman and Cowin [7], providing an alternative modeling perspective for porous and granular materials. Early mathematical investigations dealing explicitly with swelling soils in such setting include the work of Karalis [8], where elastic deformation in non-saturated swelling soils was analyzed.

Within this poroelastic viewpoint, the dynamics of swelling porous elastic soils can be written in terms of balance laws for the interacting fluid and solid constituents. In one space dimension, the basic field equations governing the displacements of the pore fluid and of the elastic solid are written as follows:

$${\rho }_1{u}_{tt}={\left(Q_1\right)}_x-I_1+E_1,{\rho }_2{\varphi }_{tt}={\left(Q_2\right)}_x+I_2+E_2,$$

(1)

where $u=u(x,t)$ and $\varphi =\varphi (x,t)$ denote, respectively, the displacement of the pore fluid constituent and the deformation of the elastic solid matrix. The constants ${\rho }_1,{\rho }_2>0$ are the corresponding mass densities, while $Q_1=Q_1(x,t)$ and $Q_2=Q_2(x,t)$ represent the partial stresses. The terms $I_i$ and $E_i$ stand for internal interaction forces and external forces acting on each constituent. The partial stresses are typically assumed to depend linearly on the strains, leading to the constitutive relations

$$Q_1=a_1{u}_x+a_2{\varphi }_x,Q_2=a_2{u}_x+a_3{\varphi }_x,$$

(2)

where the constants $a_1,a_3>0$ and $a_2\ne 0$ are elastic coefficients satisfying the structural condition

$$\begin{array}{c}\hfill a_1{a}_3>a_2^2,\end{array}$$

(3)

which ensures that the underlying elastic energy density is strictly positive or equivalently, that the associated quadratic form in $(u_x,{\varphi }_x)$ is coercive.

In the absence of internal and external forces ($I_i=E_i=0$), the evolution system (1)–(2) reduces to the classical conservative model

$$\left\{\begin{array}{c}{\rho }_1{u}_{tt}-a_1{u}_{xx}-a_2{\varphi }_{xx}=0,\hfill \\ {\rho }_2{\varphi }_{tt}-a_3{\varphi }_{xx}-a_2{u}_{xx}=0,\hfill \end{array}\right.x\in (0,L),t>0.$$

(4)

Assuming (3), this system admits the natural energy functional

$$E\left(t\right)={\displaystyle \frac{1}{2}}{\int }_0^L\left[{\rho }_1{u}_t^2+{\rho }_2{\varphi }_t^2+a_1{u}_x^2+a_3{\varphi }_x^2+2a_2{u}_x{\varphi }_x\right]dx,$$

(5)

which is positive definite and conserved along solutions, that is, $E^{\prime }\left(t\right)=0.$ In particular, the model (4) contains no intrinsic mechanism leading to decay of the mechanical energy, and stabilization cannot be expected at this level of description.

This conservative nature has, in turn, motivated a substantial amount of literature devoted to identifying dissipation mechanisms capable of producing stability in swelling porous elastic models. In particular, Quintanilla [9], in working within the balance-law formulation (1), introduced velocity-dependent interaction forces by taking

$$I_1=I_2=\xi \left(u_t-{\varphi }_t\right),E_1=a_3{u}_{xxt},E_2=0,$$

where $\xi >0$, under homogeneous Dirichlet boundary conditions, and established exponential stability of the resulting dissipative system. In a related setting, Quintanilla [10] treated swelling porous elastic soils with incompressible fluid. Under appropriate dissipative assumptions, he derived exponential decay.

Later, Wang and Guo [11] studied the same basic system under mixed boundary conditions by introducing an internal viscous damping acting on the fluid displacement. More precisely, in the balance-law formulation (1) they set

$$I_1=I_2=E_2=0,E_1=-{\rho }_1\gamma \left(x\right)u_t,$$

where $\gamma (·)\ge 0$ has positive mean. Using a spectral method, they proved exponential stability of the corresponding system. Other damping mechanisms have also been investigated, including nonlinear feedback dissipation [12], viscoelastic memory effects [13], and stabilization mechanisms involving delay terms and boundary feedback controls [14,15,16,17,18,19,20].

We now turn to swelling porous-heat dynamics in the presence of thermodiffusion effects. For background on thermodiffusion couplings in structural and continuum models, we refer, for instance, to Aouadi et al. [21]. Let $(0,L)$ denote the spatial interval occupied by the soil and let $t>0$ denote time. To account for mass diffusion and heat transfer, the balance laws (1) are coupled with additional conservation relations describing the evolution of entropy and of the diffusive concentration. More precisely, we consider the extended balance system

$${\rho }_1{u}_{tt}={\left(Q_1\right)}_x-I_1+E_1,{\rho }_2{\varphi }_{tt}={\left(Q_2\right)}_x+I_2+E_2,v_t=-q_x,C_t=-{\eta }_x,$$

(6)

where $v=v(x,t)$ denotes the entropy density, $C=C(x,t)$ denotes the concentration of the diffusive material, $q=q(x,t)$ is the heat flux, and $\eta =\eta (x,t)$ is the diffusive mass flux.

The constitutive relations are assumed to be linear and incorporate thermodiffusive coupling effects in the partial stresses and in the thermodynamic variables. In particular, we adopt

$$Q_1=a_1{u}_x+a_2{\varphi }_x,Q_2=a_2{u}_x+a_3{\varphi }_x+\gamma \theta +\beta C,v=-\gamma {\varphi }_x+\tau \theta +\rho C,p=\beta {\varphi }_x-\rho \theta +\xi C,$$

(7)

where $\theta =\theta (x,t)$ denotes the temperature variation and $p=p(x,t)$ denotes the chemical potential. The coefficients $\tau ,\gamma ,\beta >0$ are coupling parameters. The constant $\rho >0$ measures the strength of the thermodiffusion coupling between the thermal and diffusive processes, while $\xi >0$ corresponds to the purely diffusive contribution in the chemical potential.

Moreover, the fluxes are assumed to satisfy the classical Fourier and Fick laws, namely

$$\begin{array}{c}\hfill q=-k{\theta }_x,\eta =-h{p}_x,\end{array}$$

(8)

where $k>0$ is the thermal conductivity and $h>0$ is the diffusion coefficient.

In the remainder of the paper, we focus on the homogeneous case $I_i=E_i=0$ for $i=1,2$. Furthermore, instead of treating the concentration C as a primary state variable, we formulate the model in terms of the chemical potential p. Under this choice of variables, the coupled thermodiffusive system can be written as

$$\left\{\begin{array}{cc}{\rho }_1{u}_{tt}-a_1{u}_{xx}-a_2{\varphi }_{xx}=0,\hfill & (x,t)\in (0,L)\times (0,\infty ),\hfill \\ {\rho }_2{\varphi }_{tt}-b_3{\varphi }_{xx}-a_2{u}_{xx}-{\gamma }_1{\theta }_x-{\gamma }_2{p}_x=0,\hfill & (x,t)\in (0,L)\times (0,\infty ),\hfill \\ c{\theta }_t+d{p}_t-k{\theta }_{xx}-{\gamma }_1{\varphi }_{xt}=0,\hfill & (x,t)\in (0,L)\times (0,\infty ),\hfill \\ d{\theta }_t+r{p}_t-h{p}_{xx}-{\gamma }_2{\varphi }_{xt}=0,\hfill & (x,t)\in (0,L)\times (0,\infty ),\hfill \\ u(x,0)=u_0\left(x\right),u_t(t,0)=u_1\left(x\right),\hfill & x\in (0,L),\hfill \\ \varphi (x,0)={\varphi }_0\left(x\right),{\varphi }_t(t,0)={\varphi }_1\left(x\right),\hfill & x\in (0,L),\hfill \\ \theta (x,0)={\theta }_0\left(x\right),p(t,0)=p_0\left(x\right),\hfill & x\in (0,L),\hfill \\ u_x(0,t)=u(L,t)={\varphi }_x(0,t)=\varphi (L,t)=0,\hfill & t\ge 0,\hfill \\ \theta (0,t)={\theta }_x(L,t)=p(0,t)=p(L,t)=0,\hfill & t\ge 0.\hfill \end{array}\right.$$

(9)

Here the effective coefficients are given by

$$b_3=a_3-{\displaystyle \frac{{\beta }^2}{\xi }},{\gamma }_1=\gamma +{\displaystyle \frac{\rho \beta }{\xi }},{\gamma }_2={\displaystyle \frac{\beta }{\xi }},d={\displaystyle \frac{\rho }{\xi }},c=\tau +{\displaystyle \frac{{\rho }^2}{\xi }},r={\displaystyle \frac{1}{\xi }}$$

and are assumed to be strictly positive. In addition, we impose the structural conditions

$$\begin{array}{c}\hfill a_1{b}_3>a_2^2,cr>d^2,\end{array}$$

(10)

which ensure that the quadratic forms associated with the elastic part and thermodiffusive part of the energy are uniformly positive definite. Since $b_3=a_3-{\displaystyle \frac{{\beta }^2}{\xi }}<a_3$ and $a_1>0$, the inequality $a_1{b}_3>a_2^2$ implies (3), and therefore the coercivity condition in the purely mechanical setting is automatically satisfied. The functions $u_0,u_1,{\varphi }_0,{\varphi }_1,{\theta }_0,p_0$ denotes the prescribed initial data in suitable energy spaces. The boundary conditions in (9) are of mixed type and read

$$u_x(0,t)=u(L,t)={\varphi }_x(0,t)=\varphi (L,t)=\theta (0,t)={\theta }_x(L,t)=p(0,t)=p(L,t)=0,t\ge 0.$$

To the best of our knowledge, only two contributions in the current literature address close variants of the thermodiffusive swelling porous-heat model (9). Douib and Zitouni [22] considered system (9) under the action of an additional linear frictional damping term in the second equation, together with a constant delay. Under this strengthened dissipation structure, they established exponential stability without imposing the classical equal wave speed condition for the underlying hyperbolic part. Subsequently, Bouaziz et al. [23] obtained an analogous exponential stability result for a distributed delay counterpart of the same model. In both analyses, the additional frictional damping term produces a direct dissipative contribution of the form

$${\displaystyle \mathit{-}{\int }_0^L{\varphi }_t^{2}dx}$$

in the energy identity, and this dissipation plays a decisive role in establishing exponential stability of the system without imposing the equal wave speed condition.

The present work is motivated by a stability phenomenon that arises repeatedly in coupled hyperbolic–parabolic systems with thermodiffusion. In such models, the thermodiffusive coupling does introduce dissipation, yet this dissipation alone does not automatically translate into exponential stability at the energy level unless additional damping effects are incorporated or a special structural configuration is satisfied. This point becomes particularly transparent in thermodiffusive beam dynamics. As a representative example, Aouadi et al. [21] analyzed a Timoshenko system involving both thermal and mass diffusion effects, and established that, without imposing the classical equal wave speed condition, exponential stability fails in general for the Neumann problem. They further showed that the mere addition of one linear frictional damping term is sufficient to recover exponential stability for the Dirichlet case. Related parameter-sensitive stability mechanisms have since been documented for other structural models with thermodiffusion, including various Timoshenko, Bresse, and laminated beam type systems; see, for instance, refs. [24,25,26,27,28,29,30,31]. Taken together, these results indicate that exponential stabilization in thermodiffusion-driven models often hinges either on supplementary friction-type damping or on restrictive structural conditions.

In contrast to the aforementioned results, we establish exponential stability for system (9) without introducing any additional internal or boundary damping, and without imposing restrictive parameter relations such as equality of wave speeds. Our argument is carried out purely using the energy method. We also point out that the energy functional introduced in this work departs from the standard one customarily employed in the thermodiffusion literature. This modification, made precise in Lemma 1, is better adapted to the coupled structure of system (9) and avoids the appearance of non-coercive cross terms.

The remainder of the paper is organized as follows. In Section 2, we derive the fundamental energy identity and introduce the auxiliary functionals together the differential estimates they satisfy. In Section 3, these ingredients are combined to construct an appropriate Lyapunov functional, from which the exponential stability result is obtained. Throughout the paper, the letter K denotes a generic positive constant whose value may change from one line to the next; it may also absorb the Poincarés constant.

2. Technical Lemmas

This section is devoted to a set of technical lemmas that provide the differential identities and estimates needed for the proof of our exponential stability result. We introduce two positive constants that will repeatedly appear throughout the forthcoming estimates.

$$\begin{array}{c}\hfill k_1=b_3-{\displaystyle \frac{a_2^2}{a_1}}>0,k_2=r-{\displaystyle \frac{d^2}{c}}>0,\end{array}$$

(11)

which follow directly from the structural conditions $b_{1.3}{a}_1>a_2^2$ and $rc>d^2.$ Furthermore, we shall repeatedly use the following identity:

$$\begin{array}{c}\hfill u_x=\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)-{\displaystyle \frac{a_2}{a_1}}{\varphi }_x.\end{array}$$

(12)

Lemma 1. 

Let $(u,\varphi ,\theta ,p)$ be the solution of (9). Then, the modified energy functional E defined by

$$E\left(t\right)={\displaystyle \frac{1}{2}}{\int }_0^L\left[{\rho }_1{u}_t^2+{\rho }_2{\varphi }_t^2+k_1{\varphi }_x^2+k_2{p}^2+a_1{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^2+c{\left(\theta +{\displaystyle \frac{d}{c}}p\right)}^2\right]dx,$$

(13)

satisfies

$$E^{\prime }\left(t\right)=-k{\int }_0^L{\theta }_x^{2}dx-h{\int }_0^L{p}_x^{2}dx,\forall t\ge 0.$$

(14)

Proof. 

The computations carried out in this proof are standard in the stability literature, up to minor modifications reflecting the particular energy functional adopted in the present work. We test each equation in (9) against $u_t,{\varphi }_t,\theta ,$ and $p,$ respectively, and then integrate by parts while taking into account the boundary conditions. For completeness, we provide the details below. For the first equation, we have

$$\begin{array}{cc}\hfill & {\rho }_1{\int }_0^L{u}_{tt}{u}_{t}dx-a_1{\int }_0^L{u}_t\left(u_{xx}+{\displaystyle \frac{a_2}{a_1}}{\varphi }_{xx}\right)dx=0,\hfill \\ \hfill & {\displaystyle \frac{{\rho }_1}{2}}{\displaystyle \frac{d}{dt}}{\int }_0^L{u}_t^{2}dx+a_1{\int }_0^L{u}_{tx}\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx=0,\hfill \\ \hfill & {\displaystyle \frac{{\rho }_1}{2}}{\displaystyle \frac{d}{dt}}{\int }_0^L{u}_t^{2}dx+{\displaystyle \frac{a_1}{2}}{\displaystyle \frac{d}{dt}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx-a_2{\int }_0^L{\varphi }_{xt}\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx=0.\hfill \end{array}$$

(15)

For the second equation, we have

$$\begin{array}{cc}\hfill & {\rho }_2{\int }_0^L{\varphi }_{tt}{\varphi }_{t}dx+b_3{\int }_0^L{\varphi }_{xt}{\varphi }_{x}dx+a_2{\int }_0^L{\varphi }_{xt}{u}_{x}dx-{\gamma }_1{\int }_0^L{\varphi }_t{\theta }_{x}dx-{\gamma }_2{\int }_0^L{\varphi }_t{p}_{x}dx=0,\hfill \\ \hfill & {\displaystyle \frac{{\rho }_2}{2}}{\displaystyle \frac{d}{dt}}{\int }_0^L{\varphi }_t^{2}dx+a_2{\int }_0^L{\varphi }_{xt}\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx+{\displaystyle \frac{1}{2}}\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right){\displaystyle \frac{d}{dt}}{\int }_0^L{\varphi }_x^{2}dx\hfill \\ \hfill & -{\gamma }_1{\int }_0^L{\varphi }_t{\theta }_{x}dx-{\gamma }_2{\int }_0^L{\varphi }_t{p}_{x}dx=0.\hfill \end{array}$$

(16)

Following the same approach, we have for the third and fourth equations

$$\begin{array}{cc}\hfill & {\displaystyle \frac{c}{2}}{\displaystyle \frac{d}{dt}}{\int }_0^L{\left(\theta +{\displaystyle \frac{d}{c}}p\right)}^{2}dx-d{\int }_0^{L}p\left({\theta }_t+{\displaystyle \frac{d}{c}}{p}_t\right)dx+k{\int }_0^L{\theta }_x^{2}dx+{\gamma }_1{\int }_0^L{\varphi }_t{\theta }_{x}dx=0,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & d{\int }_0^{L}p\left({\theta }_t+{\displaystyle \frac{d}{c}}{p}_t\right)dx+{\displaystyle \frac{1}{2}}\left(r-{\displaystyle \frac{d^2}{c}}\right){\displaystyle \frac{d}{dt}}{\int }_0^L{p}^{2}dx+{\gamma }_2{\int }_0^L{\varphi }_t{p}_{x}dx=0,\hfill \end{array}$$

(18)

respectively. The combination of (15)–(18) gives both (13) and (14). □

Lemma 2. 

Let $(u,\varphi ,\theta ,p)$ be the solution of (9). Then, the functional defined by

$$\begin{array}{c}\hfill G_1\left(t\right)={\rho }_2{\int }_0^L\left(u+{\displaystyle \frac{a_2}{a_1}}\varphi \right){\varphi }_{t}dx-{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}{\int }_0^L{u}_t\varphi dx\end{array}$$

satisfies, for any ${\epsilon }_1>0$, the estimate

$$\begin{array}{cc}\hfill G_1^{\prime }\left(t\right)\le -& {\displaystyle \frac{a_2}{2}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\epsilon }_1{\int }_0^L{u}_t^{2}dx\hfill \\ \hfill & +K\left(1+{\displaystyle \frac{1}{{\epsilon }_1}}\right){\int }_0^L{\varphi }_t^{2}dx+K{\int }_0^L{\theta }_x^{2}dx+K{\int }_0^L{p}_x^{2}dx.\hfill \end{array}$$

(19)

Proof. 

Direct derivative of $G_1$ gives

$$\begin{array}{cc}\hfill G_1^{\prime }\left(t\right)={\rho }_2& {\int }_0^L\left(u_t+{\displaystyle \frac{a_2}{a_1}}{\varphi }_t\right){\varphi }_{t}dx+{\rho }_2{\int }_0^L\left(u+{\displaystyle \frac{a_2}{a_1}}\varphi \right){\varphi }_{tt}dx-{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}{\int }_0^L{u}_{tt}\varphi dx\hfill \\ \hfill & -{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}{\int }_0^L{u}_t{\varphi }_{t}dx,\hfill \\ \hfill =& ({\rho }_2-{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}){\int }_0^L{u}_t{\varphi }_{t}dx+{\displaystyle \frac{a_2{\rho }_2}{a_1}}{\int }_0^L{\varphi }_t^{2}dx+{\rho }_2{\int }_0^L\left(u+{\displaystyle \frac{a_2}{a_1}}\varphi \right){\varphi }_{tt}dx\hfill \\ \hfill & -{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}{\int }_0^L{u}_{tt}\varphi dx.\hfill \end{array}$$

(20)

We now use the first two equations of (9) to substitute for ${\varphi }_{tt}$ and $u_{tt}$ as follows:

$$\begin{array}{cc}\hfill {\rho }_2{\int }_0^L\left(u+{\displaystyle \frac{a_2}{a_1}}\varphi \right){\varphi }_{tt}dx=& {\int }_0^L\left(u+{\displaystyle \frac{a_2}{a_1}}\varphi \right)\left(b_3{\varphi }_{xx}+a_2{u}_{xx}+{\gamma }_1{\theta }_x+{\gamma }_2{p}_x\right)dx\hfill \\ \hfill =& -{\int }_0^L\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)\left(b_3{\varphi }_x+a_2{u}_x+{\gamma }_1\theta +{\gamma }_{2}p\right)dx.\hfill \end{array}$$

Making use of identity (12), we obtain

$$\begin{array}{cc}\hfill {\rho }_2{\int }_0^L\left(u+{\displaystyle \frac{a_2}{a_1}}\varphi \right){\varphi }_{tt}dx=-a_2& {\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx-\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right){\int }_0^L{\varphi }_x\left(u_x+{\displaystyle \frac{a_2}{a_1}}\right)dx\hfill \\ \hfill & -{\gamma }_1{\int }_0^L\theta \left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx-{\gamma }_2{\int }_0^{L}p\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx.\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill -{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}{\int }_0^L{u}_{tt}\varphi dx& =-{\displaystyle \frac{k_1}{a_1}}{\int }_0^L\varphi \left(a_1{u}_{xx}+a_2{\varphi }_{xx}\right)dx\hfill \\ \hfill & =k_1{\int }_0^L{\varphi }_x\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx.\hfill \end{array}$$

(22)

By substituting (21) and (22) into (20) and recall from (11) that $k_1=b_3-{\displaystyle \frac{a_2^2}{a_1}}$, we obtain

$$\begin{array}{cc}\hfill G_1^{\prime }\left(t\right)=-a_2& {\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\displaystyle \frac{a_2{\rho }_2}{a_1}}{\int }_0^L{\varphi }_t^{2}dx+\left({\rho }_2-{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}\right){\int }_0^L{u}_t{\varphi }_{t}dx\hfill \\ \hfill & -{\gamma }_1{\int }_0^L\theta \left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx-{\gamma }_2{\int }_0^{L}p\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx.\hfill \end{array}$$

(23)

Using Young’s inequality and Poincaré inequality with $c_p$ as the Poincaré constant, we get

$$\begin{array}{cc}\hfill \left({\rho }_2-{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}\right){\int }_0^L{u}_t{\varphi }_{t}dx& \le {\epsilon }_1{\int }_0^L{u}_t^{2}dx+{\displaystyle \frac{1}{4{\epsilon }_1}}{\left({\rho }_2-{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}\right)}^2{\int }_0^L{\varphi }_t^{2}dx,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill -{\gamma }_1{\int }_0^L\theta \left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx\le & {\displaystyle \frac{a_2}{4}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\displaystyle \frac{{\gamma }_1^2}{a_2}}{\int }_0^L{\theta }^{2}dx\hfill \\ \hfill \le & {\displaystyle \frac{a_2}{4}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\displaystyle \frac{{\gamma }_1^2{c}_p}{a_2}}{\int }_0^L{\theta }_x^{2}dx,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill -{\gamma }_2{\int }_0^{L}p\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx\le & {\displaystyle \frac{a_2}{4}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\displaystyle \frac{{\gamma }_2^2}{a_2}}{\int }_0^L{p}^{2}dx\hfill \\ \hfill \le & {\displaystyle \frac{a_2}{4}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\displaystyle \frac{{\gamma }_2^2{c}_p}{a_2}}{\int }_0^L{p}_x^{2}dx.\hfill \end{array}$$

(26)

By substituting (24)–(26) into (23), we have

$$\begin{array}{cc}\hfill G_1^{\prime }\le -{\displaystyle \frac{a_2}{2}}& {\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\epsilon }_1{\int }_0^L{u}_t^{2}dx+{\displaystyle \frac{{\gamma }_1^2}{a_2}}{\int }_0^L{\theta }^{2}dx+{\displaystyle \frac{{\gamma }_2^2{c}_p}{a_2}}{\int }_0^L{p}_x^{2}dx\hfill \\ \hfill & +\left({\displaystyle \frac{a_2{\rho }_2}{a_1}}+{\displaystyle \frac{1}{4{\epsilon }_1}}{\left({\rho }_2-{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}\right)}^2\right){\int }_0^L{\varphi }_t^{2}dx,\hfill \end{array}$$

and by absorbing the constants into K we have (19). □

Lemma 3. 

Let $(u,\varphi ,\theta ,p)$ be the solution of (9). Then, the functional defined by

$$\begin{array}{c}\hfill G_2\left(t\right)={\int }_0^L\left(c\theta +dp\right){\int }_0^x{\varphi }_t\left(y\right)dydx\end{array}$$

satisfies, for any ${\epsilon }_2,{\epsilon }_3>0$, the estimate

$$\begin{array}{cc}\hfill G_2^{\prime }\left(t\right)\le -& {\displaystyle \frac{{\gamma }_1}{2}}{\int }_0^L{\varphi }_t^{2}dx+{\epsilon }_2{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\epsilon }_3{\int }_0^L{\varphi }_x^{2}dx\hfill \\ \hfill & +K\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}+{\displaystyle \frac{1}{{\epsilon }_3}}\right){\int }_0^L{\theta }_x^{2}dx+K\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}+{\displaystyle \frac{1}{{\epsilon }_3}}\right){\int }_0^L{p}_x^{2}dx.\hfill \end{array}$$

(27)

Proof. 

Differentiating $G_2$ with respect to t, we have

$$\begin{array}{c}\hfill G_2^{\prime }\left(t\right)={\int }_0^L\left(c\theta +dp\right){\int }_0^x{\varphi }_{tt}\left(y\right)dydx+{\int }_0^L\left(c{\theta }_t+d{p}_t\right){\int }_0^x{\varphi }_t\left(y\right)dydx.\end{array}$$

(28)

Now, we consider each of the terms as follows: Using the second equation in (9) together with the boundary conditions, we obtain

$$\begin{array}{cc}\hfill {\int }_0^L\left(c\theta +dp\right){\int }_0^x{\varphi }_{tt}\left(y\right)dydx=& {\displaystyle \frac{1}{{\rho }_2}}{\int }_0^L\left(c\theta +dp\right){\int }_0^x\left[b_3{\varphi }_{yy}+a_2{u}_{yy}+{\gamma }_1{\theta }_y+{\gamma }_2{p}_y\right]dydx\hfill \\ \hfill =& {\displaystyle \frac{1}{{\rho }_2}}{\int }_0^L\left(c\theta +dp\right)\left[b_3{\varphi }_x+a_2{u}_x+{\gamma }_1\theta +{\gamma }_{2}p\right]dx\hfill \\ \hfill =& {\displaystyle \frac{c{b}_3}{{\rho }_2}}{\int }_0^L\theta {\varphi }_{x}dx+{\displaystyle \frac{c{a}_2}{{\rho }_2}}{\int }_0^L\theta u_{x}dx+{\displaystyle \frac{c{\gamma }_1}{{\rho }_2}}{\int }_0^L{\theta }^{2}dx+{\displaystyle \frac{d{\gamma }_2}{{\rho }_2}}{\int }_0^L{p}^{2}dx\hfill \\ \hfill & +\left({\displaystyle \frac{d{\gamma }_1}{{\rho }_2}}+{\displaystyle \frac{c{\gamma }_2}{{\rho }_2}}\right){\int }_0^L\theta pdx+{\displaystyle \frac{d{b}_3}{{\rho }_2}}{\int }_0^{L}p{\varphi }_{x}dx+{\displaystyle \frac{d{a}_2}{{\rho }_2}}{\int }_0^{L}p{u}_{x}dx.\hfill \end{array}$$

(29)

Similarly, using third equation in (9) together with the boundary conditions, we have

$$\begin{array}{cc}\hfill {\int }_0^L\left(c{\theta }_t+d{p}_t\right){\int }_0^x{\varphi }_t\left(y\right)dydx=& {\int }_0^L\left(k{\theta }_{xx}+{\gamma }_1{\varphi }_{xt}\right){\int }_0^x{\varphi }_t\left(y\right)dydx\hfill \\ \hfill =& -k{\int }_0^L{\theta }_x{\varphi }_{t}dx-{\gamma }_1{\int }_0^L{\varphi }_t^{2}dx.\hfill \end{array}$$

(30)

By substituting (29) and (30) into (28) and using identity (12), we obtain

$$\begin{array}{cc}\hfill G_2^{\prime }\left(t\right)=-{\gamma }_1& {\int }_0^L{\varphi }_t^{2}dx+{\displaystyle \frac{c{\gamma }_1}{{\rho }_2}}{\int }_0^L{\theta }^{2}dx+{\displaystyle \frac{d{\gamma }_2}{{\rho }_2}}{\int }_0^L{p}^{2}dx-k{\int }_0^L{\theta }_x{\varphi }_{t}dx\hfill \\ \hfill & +\left({\displaystyle \frac{d{\gamma }_1}{{\rho }_2}}+{\displaystyle \frac{c{\gamma }_2}{{\rho }_2}}\right){\int }_0^L\theta pdx+{\displaystyle \frac{c{a}_2}{{\rho }_2}}{\int }_0^L\theta \left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx\hfill \\ \hfill & +{\displaystyle \frac{d{a}_2}{{\rho }_2}}{\int }_0^{L}p\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx+{\displaystyle \frac{c}{{\rho }_2}}\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right){\int }_0^L\theta {\varphi }_{x}dx\hfill \\ \hfill & +{\displaystyle \frac{d}{{\rho }_2}}\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right){\int }_0^{L}p{\varphi }_{x}dx.\hfill \end{array}$$

(31)

Next, we estimate the last six terms in the right-hand side of (31). Applying Young’s inequality, we obtain

$$\begin{array}{cc}\hfill -k{\int }_0^L{\theta }_x{\varphi }_{t}dx\le & {\displaystyle \frac{{\gamma }_1}{2}}{\int }_0^L{\varphi }_t^{2}dx+{\displaystyle \frac{k^2}{2{\gamma }_1}}{\int }_0^L{\theta }_x^{2}dx,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{d{\gamma }_1}{{\rho }_2}}+{\displaystyle \frac{c{\gamma }_2}{{\rho }_2}}\right){\int }_0^L\theta pdx\le & {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d{\gamma }_1}{{\rho }_2}}+{\displaystyle \frac{c{\gamma }_2}{{\rho }_2}}\right){\int }_0^L{\theta }^{2}dx+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d{\gamma }_1}{{\rho }_2}}+{\displaystyle \frac{c{\gamma }_2}{{\rho }_2}}\right){\int }_0^L{p}^{2}dx,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\displaystyle \frac{c{a}_2}{{\rho }_2}}{\int }_0^L\theta \left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx\le & {\displaystyle \frac{{\epsilon }_2}{2}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\displaystyle \frac{c^2{a}_2^2}{2{\rho }_2^2{\epsilon }_2}}{\int }_0^L{\theta }^{2}dx,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{a}_2}{{\rho }_2}}{\int }_0^{L}p\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx\le & {\displaystyle \frac{{\epsilon }_2}{2}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\displaystyle \frac{d^2{a}_2^2}{2{\rho }_2^2{\epsilon }_2}}{\int }_0^L{p}^{2}dx,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\displaystyle \frac{c}{{\rho }_2}}\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right){\int }_0^L\theta {\varphi }_{x}dx\le & {\displaystyle \frac{{\epsilon }_3}{2}}{\int }_0^L{\varphi }_x^{2}dx+{\displaystyle \frac{c^2}{2{\rho }_2^2{\epsilon }_3}}{\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right)}^2{\int }_0^L{\theta }^{2}dx,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{{\rho }_2}}\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right){\int }_0^{L}p{\varphi }_{x}dx\le & {\displaystyle \frac{{\epsilon }_3}{2}}{\int }_0^L{\varphi }_x^{2}dx+{\displaystyle \frac{d^2}{2{\rho }_2^2{\epsilon }_3}}{\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right)}^2{\int }_0^L{p}^{2}dx.\hfill \end{array}$$

(37)

Substituting (32)–(37) into (31) gives

$$\begin{array}{cc}\hfill G_2^{\prime }\left(t\right)\le & -{\displaystyle \frac{{\gamma }_1}{2}}{\int }_0^L{\varphi }_t^{2}dx+{\epsilon }_2{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx+{\epsilon }_3{\int }_0^L{\varphi }_x^{2}dx+{\displaystyle \frac{k^2}{2{\gamma }_1}}{\int }_0^L{\theta }_x^{2}dx\hfill \\ \hfill & +\left({\displaystyle \frac{c{\gamma }_1}{{\rho }_2}}+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d{\gamma }_1}{{\rho }_2}}+{\displaystyle \frac{c{\gamma }_2}{{\rho }_2}}\right)+{\displaystyle \frac{c^2{a}_2^2}{2{\rho }_2^2{\epsilon }_2}}+{\displaystyle \frac{c^2}{2{\rho }_2^2{\epsilon }_3}}{\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right)}^2\right){\int }_0^L{\theta }^{2}dx\hfill \\ \hfill & +\left({\displaystyle \frac{d{\gamma }_2}{{\rho }_2}}+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d{\gamma }_1}{{\rho }_2}}+{\displaystyle \frac{c{\gamma }_2}{{\rho }_2}}\right)+{\displaystyle \frac{d^2{a}_2^2}{2{\rho }_2^2{\epsilon }_2}}+{\displaystyle \frac{d^2}{2{\rho }_2^2{\epsilon }_3}}{\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right)}^2\right){\int }_0^L{p}^{2}dx.\hfill \end{array}$$

(38)

By applying the Poincaré inequality on the second and third terms in (38), we arrive at the conclusion of the proof. □

Lemma 4. 

Let $(u,\varphi ,\theta ,p)$ be the solution of (9). Then the functional defined by

$$\begin{array}{c}\hfill G_3\left(t\right)={\rho }_2{\int }_0^L{\varphi }_t\varphi dx-{\rho }_1{\int }_0^L{u}_{t}udx\end{array}$$

satisfies the estimate

$$\begin{array}{cc}\hfill G_3^{\prime }\left(t\right)\le -& {\rho }_1{\int }_0^L{u}_t^{2}dx-{\displaystyle \frac{k_1}{2}}{\int }_0^L{\varphi }_x^{2}dx+{\rho }_2{\int }_0^L{\varphi }_t^{2}dx+K{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx\hfill \\ \hfill & +K{\int }_0^L{\theta }_x^{2}dx+K{\int }_0^L{p}_x^{2}dx.\hfill \end{array}$$

(39)

Proof. 

Differentiating $G_3$, using the first two equations of (9), and integrating by parts, we obtain

$$\begin{array}{c}\hfill G_3^{\prime }\left(t\right)={\rho }_2{\int }_0^L{\varphi }_t^{2}dx+{\rho }_2{\int }_0^L{\varphi }_{tt}\varphi dx-{\rho }_1{\int }_0^L{u}_{tt}udx-{\rho }_1{\int }_0^L{u}_t^{2}dx.\end{array}$$

(40)

We now handle the second and third terms on the right-hand side of (40). Using the second and first equations in (9), respectively, together with the boundary conditions and identity (12), we obtain

$$\begin{array}{cc}\hfill {\rho }_2{\int }_0^L{\varphi }_{tt}\varphi dx=& {\int }_0^L\left[b_3{\varphi }_{xx}+a_2{u}_{xx}+{\gamma }_1{\theta }_x+{\gamma }_2{p}_x\right]\varphi dx\hfill \\ \hfill =& -{\int }_0^L\left[\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right){\varphi }_x+a_2\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)+{\gamma }_1\theta +{\gamma }_{2}p\right]{\varphi }_{x}dx\hfill \end{array}$$

(41)

and

$$\begin{array}{cc}\hfill -{\rho }_1{\int }_0^L{u}_{tt}udx=& -{\int }_0^L\left[a_1{u}_{xx}+a_2{\varphi }_{xx}\right]udx={\int }_0^L\left[a_1{u}_x+a_2{\varphi }_x\right]u_{x}dx\hfill \\ \hfill =& a_1{\int }_0^L\left(u_x+{\displaystyle \frac{a_2^2}{a_1}}{\varphi }_x\right)u_{x}dx.\hfill \end{array}$$

(42)

By substituting (41) and (42) into (40), we obtain

$$\begin{array}{cc}\hfill G_3^{\prime }\left(t\right)=-{\rho }_1& {\int }_0^L{u}_t^{2}dx-\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right){\int }_0^L{\varphi }_x^{2}dx+{\rho }_2{\int }_0^L{\varphi }_t^{2}dx-a_2{\int }_0^L{\varphi }_x\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx\hfill \\ \hfill & +a_1{\int }_0^L{u}_x\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx-{\gamma }_1{\int }_0^L{\varphi }_x\theta dx-{\gamma }_2{\int }_0^L{\varphi }_{x}pdx.\hfill \end{array}$$

(43)

Now, we estimate the last four terms. Let ${\delta }_1>0$, by Young’s inequality, we obtain

$$\begin{array}{cc}\hfill -a_2{\int }_0^L{\varphi }_x\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx\le & {\displaystyle \frac{{\delta }_1}{4}}{\int }_0^L{\varphi }_x^{2}dx+{\displaystyle \frac{a_2^2}{{\delta }_1}}{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill a_1{\int }_0^L{u}_x\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx=& a_1{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx-a_2{\int }_0^L{\varphi }_x\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)dx\hfill \\ \hfill \le & {\displaystyle \frac{{\delta }_1}{4}}{\int }_0^L{\varphi }_x^{2}dx+\left(a_1+{\displaystyle \frac{a_2^2}{{\delta }_1}}\right){\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill -{\gamma }_1{\int }_0^L{\varphi }_x\theta dx\le & {\displaystyle \frac{{\delta }_1}{4}}{\int }_0^L{\varphi }_x^{2}dx+{\displaystyle \frac{{\gamma }_1^2}{{\delta }_1}}{\int }_0^L{\theta }^{2}dx\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill -{\gamma }_2{\int }_0^L{\varphi }_{x}pdx\le & {\displaystyle \frac{{\delta }_1}{4}}{\int }_0^L{\varphi }_x^{2}dx+{\displaystyle \frac{{\gamma }_2^2}{{\delta }_1}}{\int }_0^L{p}^{2}dx.\hfill \end{array}$$

(47)

The combination of (43)–(47) yields

$$\begin{array}{cc}\hfill G_3^{\prime }\left(t\right)\le -{\rho }_1& {\int }_0^L{u}_t^{2}dx-\left(\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right)-{\delta }_1\right){\int }_0^L{\varphi }_x^{2}dx+{\rho }_2{\int }_0^L{\varphi }_t^{2}dx+{\displaystyle \frac{{\gamma }_1^2}{{\delta }_1}}{\int }_0^L{\theta }^{2}dx\hfill \\ \hfill & +{\displaystyle \frac{{\gamma }_1^2}{{\delta }_1}}{\int }_0^L{p}^{2}dx+\left(a_1+{\displaystyle \frac{2a_2^2}{{\delta }_1}}\right){\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx.\hfill \end{array}$$

(48)

Choosing ${\delta }_1={\displaystyle \frac{k_1}{2}}={\displaystyle \frac{1}{2}}\left(b_3-{\displaystyle \frac{a_2^2}{a_1}}\right)$ and applying the Poincaré’s inequality to $\theta$ and p, we obtain (39). □

3. Exponential Stability

In this section, we establish the exponential stability of system (9).

Theorem 1. 

Let $(u,\varphi ,\theta ,p)$ be the solution of (9). Then the energy functional defined in (13) decays exponentially. More precisely, there exist positive constants $c_0$ and $c_1$ such that

$$E\left(t\right)\le c_0{e}^{-c_{1}t},\forall t\ge 0.$$

(49)

Proof. 

To establish (49), we introduce the Lyapunov functional:

$$\mathcal{L}\left(t\right):=NE\left(t\right)+N_1{G}_1\left(t\right)+N_2{G}_2\left(t\right)+4G_3\left(t\right),\forall t\ge 0,$$

(50)

where E is the total energy defined by (13) and $G_1,G_2,G_3$ are the auxiliary functionals introduced earlier in Lemma 2–4. The constants N and $N_i$ are positive real numbers to be determined below to ensure that the Lyapunov functional $\mathcal{L}$ admits a negative derivative.

Differentiating $\mathcal{L}$ and invoking the previously established estimates (14), (19), (27) and (39), we obtain

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le & -\left[kN-K{N}_1-K{N}_2\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}+{\displaystyle \frac{1}{{\epsilon }_3}}\right)-K\right]{\int }_0^L{\theta }_x^{2}dx\hfill \\ \hfill & -\left[hN-K{N}_1-K{N}_2\left(1+{\displaystyle \frac{1}{{\epsilon }_2}}+{\displaystyle \frac{1}{{\epsilon }_3}}\right)-K\right]{\int }_0^L{p}_x^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{a_2{N}_1}{2}}-{\epsilon }_2{N}_2-K\right]{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{{\gamma }_1{N}_2}{2}}-4{\rho }_2-K{N}_1\left(1+{\displaystyle \frac{1}{{\epsilon }_1}}\right)\right]{\int }_0^L{\varphi }_t^{2}dx\hfill \\ \hfill & -\left[4{\rho }_1-{\epsilon }_1{N}_1\right]{\int }_0^L{u}_t^{2}dx-\left[2k_1-{\epsilon }_3{N}_2\right]{\int }_0^L{\varphi }_x^{2}dx.\hfill \end{array}$$

Choosing the parameters

$${\epsilon }_1={\displaystyle \frac{3{\rho }_1}{N_1}},{\epsilon }_2={\displaystyle \frac{K}{N_2}},{\epsilon }_3={\displaystyle \frac{k_1}{N_2}},\mathrm{and}{N}_1={\displaystyle \frac{2(a_1+2K)}{a_2}},$$

we verify that

$${\displaystyle \frac{a_2{N}_1}{2}}-{\epsilon }_2{N}_2-K=a_1,4{\rho }_1-{\epsilon }_1{N}_1={\rho }_1,\mathrm{and}2k_1-{\epsilon }_3{N}_2=k_1,$$

and consequently, we obtain the following differential inequality:

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le & -\left[kN-K{N}_2\left(1+N_2\right)-K\right]{\int }_0^L{\theta }_x^{2}dx-{\rho }_1{\int }_0^L{u}_t^{2}dx-k_1{\int }_0^L{\varphi }_x^{2}dx\hfill \\ \hfill & -\left[hN-K{N}_2\left(1+N_2\right)-K\right]{\int }_0^L{p}_x^{2}dx-a_1{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx\hfill \\ \hfill & -\left[{\displaystyle \frac{{\gamma }_1}{2}}{N}_2-4{\rho }_2-K\right]{\int }_0^L{\varphi }_t^{2}dx.\hfill \end{array}$$

Now, by setting

$$N_2:={\displaystyle \frac{2}{{\gamma }_1}}\left(5{\rho }_2+K\right)$$

we obtain

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le & -\left[kN-K\right]{\int }_0^L{\theta }_x^{2}dx-{\rho }_1{\int }_0^L{u}_t^{2}dx-k_1{\int }_0^L{\varphi }_x^{2}dx-{\rho }_2{\int }_0^L{\varphi }_t^{2}dx\hfill \\ \hfill & -\left[hN-K\right]{\int }_0^L{p}_x^{2}dx-a_1{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx.\hfill \end{array}$$

(51)

On the other hand, from (50), we have

$$\begin{array}{cc}\hfill \left|\mathcal{L}\right(t)-NE(t\left)\right|\le & N_1|G_1\left(t\right)|+N_2|G_2\left(t\right)|+4|G_3\left(t\right)|\hfill \\ \hfill \le & N_1|{\rho }_2{\int }_0^L\left(u+{\displaystyle \frac{a_2}{a_1}}\varphi \right){\varphi }_{t}dx-{\displaystyle \frac{{\rho }_1{k}_1}{a_1}}{\int }_0^L{u}_t\varphi dx|\hfill \\ \hfill & +c{N}_2\left|{\int }_0^L\left(\theta +{\displaystyle \frac{d}{c}}p\right){\int }_0^x{\varphi }_t\left(y\right)dydx\right|+4|{\rho }_2{\int }_0^L{\varphi }_t\varphi dx-{\rho }_1{\int }_0^L{u}_{t}udx|\hfill \\ \hfill \le & K{\int }_0^L\left[u_t^2+{\varphi }_t^2+{\varphi }_x^2+p^2+{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^2+{\left(\theta +{\displaystyle \frac{d}{c}}p\right)}^2\right]dx\hfill \\ \hfill \le & KE\left(t\right),\hfill \end{array}$$

where we have used Young’s, Cauchy-Schwarz, and Poincaré inequalities. Consequently, we obtain

$$\left|\mathcal{L}\right(t)-NE(t\left)\right|\le KE\left(t\right),$$

which can be rewritten as

$$(N-K)E\left(t\right)\le \mathcal{L}\left(t\right)\le (N+K)E\left(t\right).$$

(52)

By taking N large enough such that

$${\kappa }_1:=kN-K>0,{\kappa }_2:=hN-K>0,\mathrm{and}N-K>0,$$

then (51) and (52) give

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le -{\kappa }_1& {\int }_0^L{\theta }_x^{2}dx-{\rho }_1{\int }_0^L{u}_t^{2}dx-k_1{\int }_0^L{\varphi }_x^{2}dx-{\rho }_2{\int }_0^L{\varphi }_t^{2}dx\hfill \\ \hfill & -{\kappa }_2{\int }_0^L{p}_x^{2}dx-a_1{\int }_0^L{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^{2}dx\hfill \end{array}$$

(53)

and

$$d_{1}E\left(t\right)\le \mathcal{L}\left(t\right)\le d_{2}E\left(t\right),$$

(54)

respectively, for some positive constants $d_1$ and $d_2$. Moreover, from the energy functional (13), we observe that

$$\begin{array}{cc}\hfill E\left(t\right)=& {\displaystyle \frac{1}{2}}{\int }_0^L\left[{\rho }_1{u}_t^2+{\rho }_2{\varphi }_t^2+k_1{\varphi }_x^2+k_2{p}^2+a_1{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^2+c{\left(\theta +{\displaystyle \frac{d}{c}}p\right)}^2\right]dx\hfill \\ \hfill \le & {\displaystyle \frac{1}{2}}{\int }_0^L\left[{\rho }_1{u}_t^2+{\rho }_2{\varphi }_t^2+k_1{\varphi }_x^2+c_p\left(k_2+{\displaystyle \frac{2d^2}{c}}\right)p_x^2+a_1{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^2+2c_{p}c{\theta }_x^2\right]dx.\hfill \end{array}$$

Consequently, there exists a positive constant $d_0$, such that

$$\begin{array}{c}\hfill -{\int }_0^L\left[u_t^2+{\varphi }_t^2+{\varphi }_x^2+p_x^2+{\left(u_x+{\displaystyle \frac{a_2}{a_1}}{\varphi }_x\right)}^2+{\theta }_x^2\right]dx\le -d_{0}E\left(t\right).\end{array}$$

(55)

Therefore, (53) becomes

$$\begin{array}{c}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le -c_{2}E\left(t\right),\forall t\ge 0,\end{array}$$

(56)

where $c_2$ is a positive constant. Using (54), we can rewrite (56) as

$$\begin{array}{c}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le -{\displaystyle \frac{c_2}{d_2}}\mathcal{L}\left(t\right),\forall t\ge 0.\end{array}$$

(57)

A direct integration of (57) yields

$$\begin{array}{c}\hfill \mathcal{L}\left(t\right)\le \mathcal{L}\left(0\right)e^{-{\displaystyle \frac{c_2}{d_2}}t},\forall t\ge 0.\end{array}$$

(58)

Applying (54) once more, we conclude that

$$E\left(t\right)\le {\displaystyle \frac{d_2}{d_1}}E\left(0\right)e^{-{\displaystyle \frac{c_2}{d_2}}t},\forall t\ge 0,$$

which is precisely (49) for $c_0={\displaystyle \frac{d_2}{d_1}}E\left(0\right)$ and $c_1={\displaystyle \frac{c_2}{d_2}}$. □

4. Open Problems

The results established in this work naturally suggest several directions for future investigation. We collect some of them here as open problems.

Problem 1. 

The extension of the present analysis to n-dimensional settings, $n\ge 2$, where new mathematical challenges arise due to multi-dimensional Sobolev embeddings and the more complex geometry of the boundary.

Problem 2. 

The stability analysis of the system (9) under nonlinear constitutive relations, remains, to the best of our knowledge, an open problem.

Problem 3. 

The analysis of the corresponding variable-coefficient system, in which the thermal conductivity k and the diffusion coefficient h are allowed to vary with soil saturation or porosity, rather than being assumed constant.

Problem 4. 

The development of efficient numerical schemes for system (9) and the computational validation of the exponential decay rate is also an interesting open problem.