Exponential Decay of Swelling Porous Elastic Soils with Microtemperatures Effects

Abstract

In this article, we considered the one-dimensional swelling problem in porous elastic soils with microtemperatures effects in the case of fluid saturation. First, we showed that the system is well-posed in the sens of semigroup. Then, we constructed a suitable Lyapunov functional based on the energy method and we proved that the dissipation given only by the microtemperatures is strong enough to provoke an exponential stability for the solution irrespective of the wave speeds of the system.

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1. Introduction

In [1], Eringen developed a continuum theory for a mixture consisting of three components: an elastic solid, viscous fluid, and gas. Also, the author obtained the field equations for a heat-conducting mixture. In the theory of mixtures, the great abstraction was extended by assuming that the constituents of a mixture could be modeled as superimposed continua, for that each point in the mixture was simultaneously occupied by a material point of each constituent. A brief description concerning the details of the historical development/review related to the general theory of the mixtures is given by Bedford and Drumheller in [2].

Swelling porous media have been studied in many disparate fields including soil science, hydrology, forestry, geotechnical, chemical, and mechanical engineering, and this is due to its prevalence in nature and modern technologies. In this article, we focused on the asymptotic behavior of swelling soils that belong to the porous media theory in the case of fluid saturation. Swelling soils contain clay minerals that change volume with water content changes that result in major geological hazards and extensive damage worldwide. The swelling soils are caused by the chemical attraction of water, where water molecules are incorporated in the clay structure in between the clay plates separating and destabilizing the mineral structure. The swelling clay particles have the property of forming a unit (particle) from lattice hydrated aluminum and magnesium silicate minerals. Thus, the clay’s particle is a mixture of clay platelets and adsorbed water (vicinal water). Such a particle can be thought to define a mesoscale which is large compared to platelet, but small compared to the soil itself. A proper description of the mesoscale system behavior is critical when modeling consolidation of a swelling clay soil. As pointed out by Eringen [1], this system is the prototype for diffusion type models in swelling soils ([3–5]).

As established by Ieşan [6] and simplified by Quintanilla [7] (see also [8, 9]), the basic field equations for the linear theory of swelling porous elastic soils are mathematically given by the following equation:$$\begin{array}{}\text{(1)}& {\rho }_z{z}_{tt}=H_x-P_2+F_2\text{,}{\rho }_u{u}_{tt}=T_x+P_1+F_1\text{,}\end{array}$$where the constituents $z$ and $u$ represent the displacement of the fluid and elastic solid material. The parameters ${\rho }_u$ and ${\rho }_z$ are the densities of each constituent which are assume to be strictly positive constants. $T\mathrm{a}\mathrm{n}\mathrm{d}H$ are the partial tensions, $F_1\mathrm{a}\mathrm{n}\mathrm{d}{F}_2$ are the external forces, and $P_1\mathrm{a}\mathrm{n}\mathrm{d}{P}_2$ are internal body forces associated with the dependent variables $u\mathrm{a}\mathrm{n}\mathrm{d}z$. Here, we assume that the constitutive equations of partial tensions are given as in [6] by the following equation:$$\begin{array}{}\text{(2)}& \begin{array}{c}T\\ H\end{array}=\underset{\mathcal{M}}{\underbrace{\begin{array}{cc}{\alpha }_1& {\alpha }_2\\ {\alpha }_2& {\alpha }_3\end{array}}}\begin{array}{c}{u}_x\\ z_x\end{array}\text{,}\end{array}$$where ${\alpha }_1\mathrm{a}\mathrm{n}\mathrm{d}{\alpha }_3$ are positive constants and ${\alpha }_2\ne 0$ is a real number. The matrix $\mathcal{M}$ is positive definite in the sense that$$\begin{array}{}\text{(3)}& {\alpha }_1{\alpha }_3>{\alpha }_2^2\text{.}\end{array}$$

Among the investigations that have been realized regarding the theory of swelling porous elastic soils, we cite the work of Quintanilla [7] when the author considered the following problem:$$\begin{array}{}\text{(4)}& \begin{array}{c}{\rho }_z{z}_{tt}-a_1{z}_{xx}-a_2{u}_{xx}-{\beta }_1{T}_x+\xi z_t-u_t-{\mu }_z{z}_{xxt}=0,\text{in}0,L\times 0,\infty \text{,}\\ {\rho }_u{u}_{tt}-\mu u_{xx}-a_2{z}_{xx}-{\beta }_2{T}_x-\xi z_t-u_t=0,\text{in}0,L\times 0,\infty \text{,}\\ c{T}_t-{\beta }_1{z}_{xt-}{\beta }_2{u}_{xt}-k{T}_{xx}=0,\text{in}0,L\times 0,\infty ,\end{array}\end{array}$$with the following initial data:$$\begin{array}{c}\text{(5)}& ux,0=u_{0}x,u_{t}x,0=u_{1}x,zx,0=z_{0}x\text{,}{z}_{t}x,0=z_{1}x,Tx,0=T_{0}x,x\in 0,L,\end{array}$$and the following homogeneous Dirichlet boundary conditions:$$\begin{array}{}\text{(6)}& ux,t=zx,t=Tx,t=0x=0,L\text{,}t\in 0,\infty \text{.}\end{array}$$

The author established an exponential stability result for the solution of equation (4) using the energy method in the isothermal case ($∆T=0$) and under the following condition on the following constants:$$\begin{array}{}\text{(7)}& {\beta }_1={\beta }_2=0\text{,}{a}_2^2<a_1\xi \text{,}{a}_3>0\text{.}\end{array}$$

Furthermore, in the nonisothermal case and ${\beta }_1,{\beta }_2\ne 0$, the author showed that the combination of the thermal effects with the elastic effects determines exponential stability.

In [10], Wang and Guo considered a problem of swelling of one-dimensional porous elastic soils given by the following equation:$$\begin{array}{}\text{(8)}& \begin{array}{c}{\rho }_u{u}_{tt}={\alpha }_1{u}_{xx}+{\alpha }_2{z}_{xx},\text{in}0,L\times 0,\infty \text{,}\\ {\rho }_z{z}_{tt}={\alpha }_3{z}_{xx}+{\alpha }_2{u}_{xx}+{\rho }_z\gamma x{z}_t,\text{in}0,L\times 0,\infty \text{,}\\ u0,t=u_{x}L,t=z0,t=z_{x}L,t=0,t\in 0,\infty \text{,}\\ ux,0=u_{0}x,u_{t}x,0=u_{1}x,x\in 0,L\text{,}\\ zx,0=z_{0}x,z_{t}x,0=z_{1}x,x\in 0,L\text{,}\end{array}\end{array}$$where $\gamma x$ is an internal viscous damping function satisfying the following condition:$$\begin{array}{}\text{(9)}& {\int }_0^L\gamma xdx>0\text{,}\end{array}$$and they proved that the system is exponentially stable by using the spectral method. We refer the reader to [11, 12] for some other interesting related results.

In [13], the authors considered the following system:$$\begin{array}{}\text{(10)}& \begin{array}{c}{\rho }_z{z}_{tt}-a_1{z}_{xx}-a_2{u}_{xx}=0,\text{in}0,L\times 0,\infty \text{,}\\ {\rho }_u{u}_{tt}-a_3{u}_{xx}-a_2{z}_{xx}+\gamma tg{u}_t=0,\text{in}0,L\times 0,\infty \text{,}\\ u0,t=u_{x}L,t=z0,t=z_{x}L,t=0,t\in 0,\infty \text{,}\\ ux,0=u_{0}x,u_{t}x,0=u_{1}x,x\in 0,L\text{,}\\ zx,0=z_{0}x,z_{t}x,0=z_{1}x,x\in 0,L\text{,}\end{array}\end{array}$$and under some properties of convex functions they showed that the dissipation given only by the nonlinear damping term $\gamma tg{u}_t$ is strong enough to provoke an exponential decay rate.

In [8], Apalara considered a swelling porous elastic system with a viscoelastic damping given by the following equation:$$\begin{array}{}\text{(11)}& \begin{array}{c}{\rho }_z{z}_{tt}-a_1{z}_{xx}-a_2{u}_{xx}=0,\text{in}\mathrm{0,1}\times 0,\infty \text{,}\\ {\rho }_u{u}_{tt}-a_3{u}_{xx}-a_2{z}_{xx}+{\int }_0^{t}gt-s{u}_{xx}sds=0,\text{in}\mathrm{0,1}\times 0,\infty \text{,}\\ u0,t=u1,t=z0,t=z1,t=0,t\in 0,\infty \text{,}\\ ux,0=u_{0}x,u_{t}x,0=u_{1}x,x\in \mathrm{0,1}\text{,}\\ zx,0=z_{0}x,z_{t}x,0=z_{1}x,x\in \mathrm{0,1}\text{,}\end{array}\end{array}$$where $g$ is the kernel (also known as the relaxation function) of the finite memory term. Under some assumptions on $g$, the author established a general decay result for the solution irrespective of the wave speeds of the system from which the exponential and polynomial decay results are only special cases.

Recently, in [9], the authors considered the following swelling problem in porous elastic soils with fluid saturation, viscous damping, and a time delay term.$$\begin{array}{}\text{(12)}& \begin{array}{c}{\rho }_z{z}_{tt}-a_1{z}_{xx}-a_2{u}_{xx}+{\xi }_1{z}_t+{\xi }_2{z}_{t}x,t-\tau =0,\text{in}0,L\times 0,\infty \text{,}\\ {\rho }_u{u}_{tt}-a_3{u}_{xx}-a_2{z}_{xx}=0,\text{in}0,L\times 0,\infty \text{,}\\ z0,t=z_{x}L,t=u0,t=u_{x}L,t=0,t\in 0,\infty \text{,}\\ ux,0=u_{0}x,u_{t}x,0=u_{1}x,x\in 0,L\text{,}\\ zx,0=z_{0}x,z_{t}x,0=z_{1}x,x\in 0,L\text{,}\end{array}\end{array}$$and they established an exponential decay of the solution under the appropriate assumption on the weight of the delay.

Motivated by the above work, in this article, we considered the following problem:$$\begin{array}{}\text{(13)}& \begin{array}{c}{\rho }_1{u}_{tt}={\alpha }_1{u}_{xx}+{\alpha }_2{\phi }_{xx},\text{in}0,L\times 0,\infty \text{,}\\ {\rho }_2{\phi }_{tt}={\alpha }_3{\phi }_{xx}+{\alpha }_2{u}_{xx}-k_3{w}_x,\text{in}0,L\times 0,\infty \text{,}\\ {\rho }_3{w}_t=k_1{w}_{xx}-k_{2}w-k_3{\phi }_{tx},\text{in}0,L\times 0,\infty \text{,}\\ u0,t=u_{x}L,t=\phi 0,t={\phi }_{x}L,t=0,t\in 0,\infty \text{,}\\ w_{x}0,t=wL,t=0,t\in 0,\infty \text{,}\\ ux,0=u_{0}x,u_{t}x,0=u_{1}x,\phi x,0={\phi }_{0}x,x\in 0,L\text{,}\\ {\phi }_{t}x,0={\phi }_{1}x,wx,0=w_{0}x,x\in 0,L\text{,}\end{array}\end{array}$$where $w=wx,t$ represent the microtemperature vector and the coefficients ${\alpha }_1,{\alpha }_2,{\alpha }_3,{\rho }_3,k_1,k_2,{\mathrm{a}\mathrm{n}\mathrm{d}k}_3$ are the constitutive parameters defining the coupling among the different components of the materials. By constructing a suitable Lyapunov functional which allows us to estimate the energy of the system, we showed that the unique dissipation due to the microtemperatures is strong enough to exponentially stabilize the system regardless of the wave speeds of the system. Introduction of microtemperature makes our problem different from those considered so far in the literature. The importance of the microtemperature appears in many works and this is due to the fact that many phenomena are affected by the heat present in the microvolume of bodies which is known as the microtemperature. Furthermore, the asymptotic behavior of solutions for the different types of problems is under the great influence of the microtemperature effect cited in [14–24] and the references therein.

The article is organized as follows: In Section 2, we gave the existence and uniqueness result of solutions of system (13) using some results from the semigroup theory. In Section 3, we use the multipliers method to prove the exponential stability result.

2. Well-Posedness

In this section, we gave the existence and uniqueness of solutions of system (13) using semigroup theory. First, we introduced the vector function $U={u,v,\phi ,\psi ,w}^T$, with $v=u_t$ and $\psi ={\phi }_t$. Therefore, system (13) can be rewritten as follows:$$\begin{array}{}\text{(14)}& \begin{array}{c}{U}_t+\mathcal{A}U=0,t>0\text{,}\\ Ux,0=U_{0}x={u_0,u_1,{\phi }_0,{\phi }_1,w_0}^T\text{,}\end{array}\end{array}$$where the operator $\mathcal{A}$ is defined by the following equation:$$\begin{array}{}\text{(15)}& \mathcal{A}=\begin{array}{ccccc}\begin{array}{c}0\end{array}& \begin{array}{c}-I\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}0\end{array}\\ \begin{array}{c}-\frac{{\alpha }_1}{{\rho }_1}{\partial }_{xx}\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}-\frac{{\alpha }_2}{{\rho }_1}{\partial }_{xx}\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}0\end{array}\\ \begin{array}{c}0\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}-I\end{array}& \begin{array}{c}0\end{array}\\ \begin{array}{c}-\frac{{\alpha }_2}{{\rho }_2}{\partial }_{xx}\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}-\frac{{\alpha }_3}{{\rho }_2}{\partial }_{xx}\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}\frac{k_3}{{\rho }_2}{\partial }_x\end{array}\\ \begin{array}{c}0\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}0\end{array}& \begin{array}{c}\frac{{\alpha }_3}{{\rho }_3}{\partial }_x\end{array}& \begin{array}{c}\frac{1}{{\rho }_3}-k_1{\partial }_{xx}+k_{2}I\end{array}\end{array}\text{.}\end{array}$$

We considered the following spaces:$$\begin{array}{c}\text{(16)}& {\tilde{H}}^{1}0,L=f\in H^{1}0,L;f0=0\text{,}{\tilde{H}}^{2}0,L=f\in H^{2}0,L;f_{x}0=fL=0\text{,}\\ {\tilde{H}}_{\ast }^{2}0,L=f\in H^{2}0,L;f0=f_{x}L=0\text{,}\\ \mathcal{H}={\tilde{H}}^{1}0,L\times L^{2}0,L\times {\tilde{H}}^{1}0,L\times L^{2}0,L\times L^{2}0,L\text{.}\end{array}$$

Then, $\mathcal{H}$, along with the inner product$$\begin{array}{c}\text{(17)}& {U,\tilde{U}}_{\mathcal{H}}={\rho }_1{\int }_0^{L}v\tilde{v}dx+{\rho }_2{\int }_0^L\psi \tilde{\psi }dx+{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x{\tilde{\phi }}_{x}dx{\int }_0^L\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\tilde{\phi }}_x+\sqrt{{\alpha }_1}{\tilde{u}}_{x}dx+{\rho }_3{\int }_0^{L}w\tilde{w}dx\text{,}\end{array}$$is a Hilbert space for any $U={u,v,\phi ,\psi ,w}^T\in \mathcal{H}$ and $\tilde{U}={\tilde{u},\tilde{v},\tilde{\phi },\tilde{\psi },\tilde{w}}^T\in \mathcal{H}$. The domain of $\mathcal{A}$ is given by the following equation:$$\begin{array}{}\text{(18)}& D\mathcal{A}=U\in \mathcal{H}u\in {\tilde{H}}_{\ast }^{2}0,L;v\in {\tilde{H}}^{1}0,L;\phi \in {\tilde{H}}_{\ast }^{2}0,L;\psi \in {\tilde{H}}^{1}0,L;w\in {\tilde{H}}^{2}0,L\text{.}\end{array}$$

It is easy to see that the operator $\mathcal{A}$ is maximal-monotone in the energy space $\mathcal{H}$. Then, by the Lumer–Phillips Theorem ([25], Theorem 4.3), we can conclude that $\mathcal{A}$ is the infinitesimal generator of $C_0$-semigroup of contraction. We are now in a position to state the following result:

Theorem 1.

Let $U_0\in \mathcal{H}$ and assume that equation (3) holds. Then, there exists a unique solution $U\in C{\mathbb{R}}_{+},\mathcal{H}$ for system (13). Moreover, if $U_0\in D\mathcal{A}$, then$$\begin{array}{}\text{(19)}& U\in C{\mathbb{R}}_{+},D\mathcal{A}\cap C^1{\mathbb{R}}_{+},\mathcal{H}\text{.}\end{array}$$

3. Exponential Decay

In this section, we stated and proved that technical lemmas are needed for the proof of our stability result.

Lemma 2.

Let $u,\phi ,w$ be a solution of system (13). Then, the energy functional $Et$ is defined by the following equation:$$\begin{array}{}\text{(20)}& Et=\frac{{\rho }_1}{2}{\int }_0^L{u}_t^{2}dx+\frac{{\rho }_2}{2}{\int }_0^L{\phi }_t^{2}dx+\frac{1}{2}{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx+\frac{1}{2}{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx+{\rho }_3{\int }_0^L{w}^{2}dx\text{,}\end{array}$$which satisfies the following equation:$$\begin{array}{}\text{(21)}& E^{\prime }t=-k_2{\int }_0^L{w}^{2}dx-k_1{\int }_0^L{w}_x^{2}dx\le 0\text{.}\end{array}$$

Proof.

Multiplying systems (13)₁, (13)₂, and (13)₃ by $u_t$, ${\phi }_t$, and $w$ respectively, integrating over $0,L$, taking into account the boundary conditions and summing them up, we obtain the following equation:$$\begin{array}{c}\text{(22)}& \frac{d}{2dt}{\int }_0^L{\rho }_1{u}_t^2+{\rho }_2{\phi }_t^2+{\rho }_3{w}^2+{\alpha }_3{\phi }_x^2+{\alpha }_1{u}_x^2+2{\alpha }_2{\phi }_x{u}_x=-k_2{\int }_0^L{w}_x^{2}dx-k_3{\int }_0^L{w}^{2}dx\text{.}\end{array}$$

Using the fact that$$\begin{array}{c}\text{(23)}& {\alpha }_3{\phi }_x^2+{\alpha }_1{u}_x^2+2{\alpha }_2{\phi }_x{u}_x={\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx+{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx\text{.}\end{array}$$

Inserting (23) in (22), we get (20) and (21).

Lemma 3.

Let $u,\phi ,w$ be a solution of system (13). Then, the functional$$\begin{array}{}\text{(24)}& I_{1}t={\rho }_2{\int }_0^L{\phi }_t\phi dx-\frac{{\alpha }_2}{{\alpha }_1}{\rho }_1{\int }_0^L{u}_t\phi dx,\hspace{1em}t\ge 0\text{,}\end{array}$$satisfies, for any ${\epsilon }_1>0$, the following equation:$$\begin{array}{}\text{(25)}& I_1^{\prime }t\le -\frac{1}{2}{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx+{\rho }_2+\frac{{\alpha }_2^2{\rho }_1^2}{4{\epsilon }_1{\alpha }_1^2}{\int }_0^L{\phi }_t^{2}dx+{\epsilon }_1{\int }_0^L{u}_t^{2}dx+C_0{\int }_0^L{w}^{2}dx,\hspace{1em}\forall t\ge 0\text{.}\end{array}$$

Proof.

By differentiating $I_{1}t$ using systems (13)₁ and (13)₂ and integrating by parts together with the boundary conditions, we obtain the following equation:$$\begin{array}{}\text{(26)}& I_1^{\prime }t=-{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx+{\rho }_2{\int }_0^L{\phi }_t^{2}dx+k_3{\int }_0^L{\phi }_{x}wdx-\frac{{\alpha }_2}{{\alpha }_1}{\rho }_1{\int }_0^L{u}_t{\phi }_{t}dx\text{.}\end{array}$$

Young’s inequality leads to the following equations:$$\begin{array}{}\text{(27)}& k_3{\int }_0^L{\phi }_{x}wdx\le \frac{1}{2}{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx+\frac{k_3^2{\alpha }_1}{2{\alpha }_1{\alpha }_3-{\alpha }_2^2}{\int }_0^L{w}^{2}dx\text{,}\text{(28)}& -\frac{{\alpha }_2}{{\alpha }_1}{\rho }_1{\int }_0^L{u}_t{\phi }_{t}dx\le {\epsilon }_1{\int }_0^L{u}_t^{2}dx+\frac{{\alpha }_2^2{\rho }_1^2}{4{\epsilon }_1{\alpha }_1^2}{\int }_0^L{\phi }_t^{2}dx\text{.}\end{array}$$

Substituting (27) and (28) in (26), we get (25).

Lemma 4.

Let $u,\phi ,w$ be a solution of system (13). Then, the functional$$\begin{array}{}\text{(29)}& I_{2}t={\rho }_2{\alpha }_2{\int }_0^L{\phi }_t\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}\phi +\sqrt{{\alpha }_1}udx-\frac{{\alpha }_2^2}{{\alpha }_1}{\rho }_1{\int }_0^L{u}_t\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}\phi +\sqrt{{\alpha }_1}udx,\hspace{1em}t\ge 0\text{,}\end{array}$$satisfies, for any ${\epsilon }_2>0$, the following equation:$$\begin{array}{}\text{(30)}& I_2^{\prime }t\le -\frac{{\alpha }_2^2{\rho }_1}{2\sqrt{{\alpha }_1}}{\int }_0^L{u}_t^{2}dx+C_1{\int }_0^L{\phi }_t^{2}dx+C_{{\epsilon }_2}{\int }_0^L{\phi }_x^{2}dx+{\epsilon }_2{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx+C_{{\epsilon }_2}{\int }_0^L{w}^2+{\phi }_x^{2}dx\text{.}\end{array}$$

Proof.

By differentiating $I_{2}t$ using systems (13)₁ and (13)₂ and integrating by parts together with the boundary conditions, we obtain the following equation:$$\begin{array}{c}\text{(31)}& I_2^{\prime }t=-\frac{{\alpha }_2^2{\rho }_1}{\sqrt{{\alpha }_1}}{\int }_0^L{u}_t^{2}dx+{\alpha }_2{\rho }_2\sqrt{{\alpha }_1}-\frac{{\alpha }_2^2{\rho }_1}{{\alpha }_1\sqrt{{\alpha }_1}}{\int }_0^L{u}_t{\phi }_{t}dx-{\alpha }_2{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x{\phi }_{x}dx+\frac{{\alpha }_2^2{\rho }_2}{\sqrt{{\alpha }_1}}{\int }_0^L{\phi }_t^{2}dx\\ +{\alpha }_2{k}_3{\int }_0^L\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_{x}wdx\text{.}\end{array}$$

Using Young’s inequality, we get the following equations:$$\begin{array}{c}\text{(32)}& {\alpha }_2{\rho }_2\sqrt{{\alpha }_1}-\frac{{\alpha }_2^2{\rho }_1}{{\alpha }_1\sqrt{{\alpha }_1}}{\int }_0^L{u}_t{\phi }_{t}dx\le \frac{{\alpha }_2^2{\rho }_1}{2\sqrt{{\alpha }_1}}{\int }_0^L{u}_t^{2}dx+C_1{\int }_0^L{\phi }_t^{2}dx\text{,}\text{(33)}& -{\alpha }_2{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x{\phi }_{x}dx\le \frac{{\epsilon }_2}{2}{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx+C_{{\epsilon }_2}{\int }_0^L{\phi }_x^{2}dx\text{,}\\ \text{(34)}& {\alpha }_2{k}_3{\int }_0^L\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_{x}wdx\le \frac{{\epsilon }_2}{2}{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx+C_{{\epsilon }_2}{\int }_0^L{w}^{2}dx\text{.}\end{array}$$

Inserting (32)–(34) in (31), we obtain (30).

Lemma 5.

Let $u,\phi ,w$ be a solution of system (13). Then, the functional$$\begin{array}{}\text{(35)}& I_{3}t={\rho }_2{\int }_0^L\phi {\phi }_{t}dx+{\rho }_1{\int }_0^{L}u{u}_{t}dx,\hspace{1em}t\begin{array}{c}\ge \end{array}0\text{,}\end{array}$$satisfies the following equation:$$\begin{array}{}\text{(36)}& I_3^{\prime }t\le -{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx-\frac{1}{2}{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx+{\rho }_2{\int }_0^L{\phi }_t^{2}dx+{\rho }_1{\int }_0^L{u}_t^{2}dx+C_2{\int }_0^L{w}^{2}dx\text{.}\end{array}$$

Proof.

By exploiting the functional $I_{4}t$ using systems (13)₄ and (13)₂ and integrating by parts, we obtain the following equation:$$\begin{array}{}\text{(37)}& I_3^{\prime }t={\rho }_2{\int }_0^L{\phi }_t^{2}dx+{\rho }_1{\int }_0^L{u}_t^{2}dx-{\alpha }_3{\int }_0^L{\phi }_x^{2}dx-2{\alpha }_2{\int }_0^L{u}_x{\phi }_{x}dx-{\alpha }_1{\int }_0^L{u}_x^{2}dx+k_3{\int }_0^{L}w{\phi }_{x}dx\text{.}\end{array}$$

Note that$$\begin{array}{c}\text{(38)}& -{\alpha }_3{\int }_0^L{\phi }_x^{2}dx-2{\alpha }_2{\int }_0^L{u}_x{\phi }_{x}dx-{\alpha }_1{\int }_0^L{u}_x^{2}dx=-{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx-{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx\text{.}\end{array}$$

So, equation (37) becomes as follows:$$\begin{array}{}\text{(39)}& I_3^{\prime }t={\rho }_2{\int }_0^L{\phi }_t^{2}dx+{\rho }_1{\int }_0^L{u}_t^{2}dx+k_3{\int }_0^{L}w{\phi }_{x}dx-{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx-{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx\text{.}\end{array}$$

Using Young’s inequality,$$\begin{array}{}\text{(40)}& k_3{\int }_0^{L}w{\phi }_{x}dx\le \frac{1}{2}{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^L{\phi }_x^{2}dx+C_2{\int }_0^L{w}^{2}dx\text{.}\end{array}$$

Substituting (40) into (39), we get (36).

Lemma 6.

Let $u,\phi ,w$ be a solution of system (13). Then, the functional$$\begin{array}{}\text{(41)}& I_{4}t={\rho }_3{\rho }_2{\int }_0^{L}w{\int }_x^L{\phi }_{t}ydydx,\hspace{1em}\forall t\ge 0\text{,}\end{array}$$satisfies, for any ${\epsilon }_3,{\epsilon }_4>0$, the following estimate:$$\begin{array}{}\text{(42)}& I_4^{\prime }t\le -\frac{k_3{\rho }_2}{2}{\int }_0^L{\phi }_t^{2}dx+{\epsilon }_3{\int }_0^L{\phi }_x^{2}dx+{\epsilon }_4{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx+C_3{\int }_0^1{w}_x^{2}dx+C_{3}1+\frac{1}{{\epsilon }_3}+\frac{1}{{\epsilon }_4}{\int }_0^1{w}^{2}dx\text{.}\end{array}$$

Proof.

By differentiating $I_{4}t$, using systems (13)₂ and (13)₃ and integrating by parts, we obtain the following equation:$$\begin{array}{}\text{(43)}& I_4^{\prime }t=k_1{\rho }_2{\int }_0^L{w}_x{\phi }_{t}dx-k_3{\rho }_2{\int }_0^L{\phi }_t^{2}dx-k_2{\rho }_2{\int }_0^{L}w{\int }_x^L{\phi }_{t}ydydx-{\alpha }_3{\rho }_3{\int }_0^{L}w{\phi }_{x}dx-{\alpha }_2{\rho }_3{\int }_0^{L}w{u}_{x}dx+{\rho }_3{k}_3{\int }_0^L{w}^{2}dx\text{.}\end{array}$$

Using the fact that$$\begin{array}{c}\text{(44)}& -{\alpha }_3{\rho }_3{\int }_0^{L}w{\phi }_{x}dx-{\alpha }_2{\rho }_3{\int }_0^{L}w{u}_{x}dx=-{\rho }_3{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^{L}w{\phi }_{x}dx-\frac{{\rho }_3{\alpha }_2}{\sqrt{{\alpha }_1}}{\int }_0^{L}w\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_{x}dx\text{.}\end{array}$$

Then, equation (43) can be rewritten as follows:$$\begin{array}{}\text{(45)}& I_4^{\prime }t=k_1{\rho }_2{\int }_0^L{w}_x{\phi }_{t}dx-k_3{\rho }_2{\int }_0^L{\phi }_t^{2}dx-k_2{\rho }_2{\int }_0^{L}w{\int }_x^L{\phi }_{t}ydydx-{\rho }_3{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^{L}w{\phi }_{x}dx-\frac{{\rho }_3{\alpha }_2}{\sqrt{{\alpha }_1}}{\int }_0^{L}w\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_{x}dx\text{.}\end{array}$$

Young’s inequality leads to the following equations:$$\begin{array}{c}\text{(46)}& -{\rho }_3{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{\int }_0^{L}w{\phi }_{x}dx\le {\epsilon }_3{\int }_0^L{\phi }_x^{2}dx+\frac{C_3}{{\epsilon }_3}{\int }_0^L{w}^{2}dx\text{,}\text{(47)}& -\frac{{\rho }_3{\alpha }_2}{\sqrt{{\alpha }_1}}{\int }_0^{L}w\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_{x}dx\le {\epsilon }_4{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx+\frac{C_3}{{\epsilon }_4}{\int }_0^L{w}^{2}dx\text{,}\\ \text{(48)}& k_1{\rho }_2{\int }_0^L{w}_x{\phi }_{t}dx\le \frac{k_3{\rho }_2}{4}{\int }_0^L{\phi }_t^{2}dx+C_3{\int }_0^L{w}_x^{2}dx\text{.}\end{array}$$

Using Young’s and Cauchy Schwarz inequalities, we find$$\begin{array}{}\text{(49)}& -k_2{\rho }_2{\int }_0^{L}w{\int }_x^L{\phi }_{t}ydydx\le \frac{k_3{\rho }_2}{4}{\int }_0^L{\phi }_t^{2}dx+C_3{\int }_0^L{w}^{2}dx\text{.}\end{array}$$

Inserting (46)–(49) into (45), we obtain (42).

Now, we define the Lyapunov functional $\mathcal{L}t$ by the following equation:$$\begin{array}{}\text{(50)}& \mathcal{L}t=NEt+N_1{I}_{1}t+N_2{I}_{2}t+2I_{3}t+N_3{I}_{4}t,\end{array}$$where $N$, $N_1$, $N_2$, and $N_3$ are positive constants.

Theorem 7.

Let $u,\phi ,w$ be a solution of system (13). Then, there exist two positive constants ${\kappa }_1$ and ${\kappa }_2$ such that the Lyapunov functional (50) satisfies the following equations:$$\begin{array}{}\text{(51)}& {\kappa }_{1}Et\le \mathcal{L}t\le {\kappa }_{2}Et,\hspace{1em}\forall t\ge 0\text{,}\text{(52)}& {\mathcal{L}}^{\prime }t\le -{\beta }_{1}Et\text{.}\end{array}$$

Proof.

From (50), we have the following equation:$$\begin{array}{c}\text{(53)}& \mathcal{L}t-NEt\le N_1{\rho }_2{\int }_0^L{\phi }_t\phi dx+N_1\frac{{\alpha }_2}{{\alpha }_1}{\rho }_1{\int }_0^L{u}_t\phi dx+N_2{\rho }_2{\alpha }_2{\int }_0^L{\phi }_t\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}\phi +\sqrt{{\alpha }_1}udx\\ +N_2\frac{{\alpha }_2^2}{{\alpha }_1}{\rho }_1{\int }_0^L{u}_t\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}\phi +\sqrt{{\alpha }_1}udx\\ +2{\rho }_2{\int }_0^L\phi {\phi }_{t}dx+2{\rho }_1{\int }_0^{L}u{u}_{t}dx\\ +N_{3}c{\rho }_2{\int }_0^{L}w{\int }_x^L{\phi }_{t}ydydx\text{.}\end{array}$$

By using the Young’s, Poincaré, Cauchy–Schwarz inequalities, we obtain the following equation:$$\begin{array}{}\text{(54)}& \mathcal{L}t-NEt\le \tau Et\text{,}\end{array}$$which yields the following equation:$$\begin{array}{}\text{(55)}& N-\tau Et\le \mathcal{L}t\le N+\tau Et\text{,}\end{array}$$and by choosing $N$ (depending on $N_1$, $N_2$ and $N_3$) sufficiently large, we obtain equations (51). Now, By differentiating $\mathcal{L}t$, exploiting equations (21), (25), (30), (36) and (42) and setting ${\epsilon }_1=1/N_1$, ${\epsilon }_2=1/2N_2$, ${\epsilon }_3=1/N_3$, and ${\epsilon }_4=1/2N_3$, we get the following equation:$$\begin{array}{c}\text{(56)}& {\mathcal{L}}^{\prime }t\le -\frac{k_3{\rho }_2}{2}{N}_3-N_1{\rho }_2+N_1\frac{{\alpha }_2^2{\rho }_1^2}{4{\alpha }_1^2}-N_2{C}_1-2{\rho }_2{\int }_0^L{\phi }_t^{2}dx-\frac{1}{2}{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{N}_1+2-C_{{\epsilon }_2}{N}_2-1{\int }_0^L{\phi }_x^{2}dx\\ -\frac{{\alpha }_2^2{\rho }_1}{2\sqrt{{\alpha }_1}}{N}_2-1-2{\rho }_1{\int }_0^L{u}_t^{2}dx-{\int }_0^L{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^{2}dx\\ -N{k}_2-N_3{C}_{3}1+N_3+2N_3-2C_2-C_{{\epsilon }_2}{N}_2-N_1{C}_0{\int }_0^L{w}^{2}dx\\ -N{k}_1-C_3{N}_3{\int }_0^L{w}_x^{2}dx.\end{array}$$

Now, we select our parameters appropriately as follows:

First, we choose $N_2$ large enough so that$$\begin{array}{}\text{(57)}& \frac{{\alpha }_2^2{\rho }_1}{2\sqrt{{\alpha }_1}}{N}_2-1-2{\rho }_1>0\text{.}\end{array}$$

Next, we select $N_1$ large enough so that$$\begin{array}{}\text{(58)}& \frac{1}{2}{\alpha }_3-\frac{{\alpha }_2^2}{{\alpha }_1}{N}_1+2-C_{{\epsilon }_2}{N}_2-1>0\text{.}\end{array}$$

We take $N_3$ large such that$$\begin{array}{}\text{(59)}& \frac{k_3{\rho }_2}{2}{N}_3-N_1{\rho }_2+N_1\frac{{\alpha }_2^2{\rho }_1^2}{4{\alpha }_1^2}-N_2{C}_1-2{\rho }_2>0\text{.}\end{array}$$

Finally, we choose $N$ large enough so that equation (51) remains valid; furthermore,$$\begin{array}{}\text{(60)}& \begin{array}{c}N{k}_2-N_3{C}_{3}1+N_3+2N_3-2C_2-C_{{\epsilon }_2}{N}_2-N_1{C}_0>0\text{,}\\ \text{and}\\ N{k}_1-C_3{N}_3>0\text{.}\end{array}\end{array}$$

All these choices with relation (56) lead to the following equation:$$\begin{array}{}\text{(61)}& {\mathcal{L}}^{\prime }t\le -{\alpha }_1{\int }_0^L{u}_t^2+{\phi }_x^2+{\phi }_t^2+{\frac{{\alpha }_2}{\sqrt{{\alpha }_1}}{\phi }_x+\sqrt{{\alpha }_1}{u}_x}^2+w^{2}dx\text{.}\end{array}$$

On the other hand, from equation (20), we obtain equation (52).

Now, we can state and prove the following stability result.

Lemma 8.

Let $u,\phi ,w$ be a solution of system (13). Then, for any $U_0\in D\mathcal{A}$, there exist two positive constants ${\lambda }_1$ and ${\lambda }_2$ such that$$\begin{array}{}\text{(62)}& Et\le {\lambda }_2{e}^{-{\lambda }_{1}t},\hspace{1em}\forall t\ge 0\text{.}\end{array}$$

Proof.

By using estimation (52), we get the following equation:$$\begin{array}{}\text{(63)}& {\mathcal{L}}^{\prime }t\le -{\beta }_{1}Et,\hspace{1em}t\ge 0\text{,}\end{array}$$having in mind the equivalence of $Et$ and $\mathcal{L}t$, we infer that$$\begin{array}{}\text{(64)}& {\mathcal{L}}^{\prime }t\le -{\lambda }_1\mathcal{L}t,\hspace{1em}t\ge 0\text{,}\end{array}$$where ${\lambda }_1={\beta }_1/{\kappa }_2$. A simple integration of equation (64) gives the following equation:$$\begin{array}{}\text{(65)}& \mathcal{L}t\le \mathcal{L}0e^{-{\lambda }_{1}t},\hspace{1em}t\ge 0\text{,}\end{array}$$which yields the serial result (62) by using the other side of the equivalence relation (51) again. The proof is complete.

Acknowledgments

This research has been funded by the Deputy for Research & Innovation, Ministry of Education through Initiative of Institutional Funding at University of Ha’il, Saudi Arabia, through project number IFP-22 021.

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