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Global existence and exponential stability of coupled Lamé system with distributed delay and source term without memory term

Abstract

In this paper, we prove the global existence and exponential energy decay results of a coupled Lamé system with distributed time delay, nonlinear source term, and without memory term by using the Faedo–Galerkin method. In addition, an appropriate Lyapunov functional, more general relaxation functions, and some properties of convex functions are considered.

Introduction

In this work, we study the following Lamé system in \(\Omega \times \mathbb{R}^{+}\):

$$\begin{aligned} \textstyle\begin{cases} u_{tt}-\Delta _{e}u+k_{1} u_{t}+\int _{\tau _{1}}^{ \tau _{2}}\mu _{1}(\varrho ) u_{t}(x,t-\varrho )\,d\varrho =f_{1} ( u,v ), \\ v_{tt}-\Delta _{e}v+k_{2} v_{t}+\int _{\tau _{1}}^{\tau _{2}}\mu _{2}( \varrho ) v_{t}(x,t-\varrho )\,d\varrho =f_{2} ( u,v ), \\ u(x,t)=v(x,t)=0 \quad \text{on } \partial \Omega \times \mathbb{R}^{+}, \\ u(x,0)=u_{0}(x),\qquad v(x,0)=v_{0}(x),\qquad u_{t}(x,0)=u_{1}(x),\\ v_{t}(x,0)=v_{1}(x),\quad x \in \Omega, \\ (u_{t}(x,-t),v_{t}(x,-t))=(f_{0}(x,t),g_{0}(x,t))\quad \text{in }\Omega \times (0,\tau _{2}), \end{cases}\displaystyle \end{aligned}$$
(1.1)

where Ω is a bounded domain in \(\mathbb{R}^{n}\) \((n=1,2,3)\) with smooth boundary Ω. The elasticity differential operator \(\Delta _{e}\) is given by

$$\begin{aligned} \Delta _{e}u=\mu \Delta u+ (\mu +\lambda ) \nabla (\operatorname{div} u ), \end{aligned}$$

and the Lamé constants μ and λ satisfy the following conditions:

$$\begin{aligned} \mu >0,\qquad \mu +\lambda >0. \end{aligned}$$

The parameters \(k_{1}, k_{2}, \tau _{1}\), and \(\tau _{2} \) are positive constants, with \(\tau _{1}<\tau _{2} \). The functions \(\mu _{1}, \mu _{2}: [\tau _{1},\tau _{2} ]\rightarrow \mathbb{R} \) are bounded. The functions \(f_{1} ( u,v )\) and \(f_{2} ( u,v )\), which represent the source terms, will be specified later.

After several authors have studied the problems of coupled systems and hyperbolic systems, their stability is associated with velocities and is proven under some given conditions (see, for example, [111]). In recent years, several authors have been interested in studying the existence and stability for Lamé systems, we refer to [1214]. The Lamé system with localized nonlinear damping and a general decay result of energy have been considered by some recent works (see, for example, [12, 14], and [15]). Bchatnia et al. in [16] investigated Lamé systems with past history. Then, Taouaf et al. in [17] established the well-posedness and asymptotic stability for the Lamé system with internal distributed delay.

Beniani et al. [13] proved the well-posedness and exponential stability of the following coupled Lamé system:

$$\begin{aligned} \textstyle\begin{cases} u_{tt}+\alpha v-\Delta _{e}u+\int _{0}^{t}g_{1} ( t-s ) \Delta u ( s ) \,ds-\mu _{1}\Delta u_{t} =0, \\ v_{tt}+\alpha u-\Delta _{e}v+\int _{0}^{t}g_{2} ( t-s ) \Delta v ( s ) \,ds-\mu _{2}\Delta v_{t}=0. \end{cases}\displaystyle \end{aligned}$$
(1.2)

After that, Baowei et al. in [18] considered the same problem with more general assumption on the relaxation functions. They established an explicit and general decay result, which are optimal, to the system.

Boulaaras et al. in [14] considered the previous problem with a source term, where under some suitable conditions on the initial data and the relaxation functions, they proved an asymptotic stability result of global solution taking into account that the kernel is not necessarily decreasing.

In the present work, we prove the existence and general decay results of problem (1.1) with respect to the presence of distributed term delay in order to ensure fast stability under some given conditions. We establish the exponential energy decay results to the system by using an appropriate Lyapunov functional.

This paper is organized as follows. In the second section, we give some preliminaries related to problem (1.1). In Sect. 3, we prove the global existence by using Faedo–Galerkin method. In the fourth section, we prove our main result of exponential energy decay.

Preliminaries

In this section, we provide some materials and necessary assumptions which we need in the proof of our results. We use the standard Lebesgue and Sobolev spaces with their scaler products and norms. For simplicity, we would write \(\Vert {\cdot} \Vert \) instead of \(\Vert {\cdot} \Vert _{2}\).

(A1) For the source terms \(f_{1} \) and \(f_{2} \), we take

$$\begin{aligned} f_{1}(u,v)=\alpha \vert u+v \vert ^{p-1}(u+v)+\beta \vert u \vert ^{\frac{p-3}{2}}u \vert v \vert ^{ \frac{p+1}{2}}, \quad\forall (u,v) \in {} \bigl[ \mathbb{R}^{n} \bigr]^{2} \end{aligned}$$

and

$$\begin{aligned} f_{2}(u,v)=\alpha \vert u+v \vert ^{p-1}(u+v)+\beta \vert v \vert ^{\frac{p-3}{2}}v \vert u \vert ^{ \frac{p+1}{2}}, \quad\forall (u,v) \in {} \bigl[ \mathbb{R}^{n} \bigr]^{2}, \end{aligned}$$

with \(\alpha,\beta >0 \). Clearly,

$$\begin{aligned} uf_{1} ( u,v ) +vf_{2} ( u,v )= ( p+1 ) F ( u,v ),\quad \forall (u,v)\in {} \bigl[ \mathbb{R}^{n} \bigr]^{2}, \end{aligned}$$
(2.1)

where

$$\begin{aligned} F ( u,v )=\frac{1}{ ( p+1 )} \bigl[ \alpha \vert u+v \vert ^{p+1}+2 \beta \vert uv \vert ^{\frac{p+1}{2}} \bigr],\quad \forall (u,v)\in {} \bigl[ \mathbb{R}^{n} \bigr]^{2}, \end{aligned}$$
(2.2)

and

$$\begin{aligned} f_{1} ( u,v )=\frac{\partial F}{\partial u},\qquad f_{2} ( u,v )= \frac{\partial F}{\partial v}. \end{aligned}$$
(2.3)

(A2)

$$\begin{aligned} \text{If } n=1,2; \quad p\geq 3, \qquad\text{if } n=3;\quad p=3. \end{aligned}$$
(2.4)

So, we have the embedding

$$\begin{aligned} H_{0}^{1} ( \Omega ) \hookrightarrow L^{q} ( \Omega )\quad \text{for } 2\leq q \leq \frac{2n}{n-2} \text{ if } n\geq 3 \quad\text{or}\quad q\geq 2 \quad\text{if } n=1,2 \end{aligned}$$

and

$$\begin{aligned} L^{r}\hookrightarrow L^{q} \quad\text{for } q< r. \end{aligned}$$

Let \(c_{s}\) be the same embedding constant, so we have

$$\begin{aligned} \Vert \nu \Vert _{q}\leq c_{s} \Vert \nabla \nu \Vert _{2},\qquad \Vert \nu \Vert _{q}\leq c_{s} \Vert \nu \Vert _{r} \quad\text{for } \nu \in H_{0}^{1} ( \Omega ). \end{aligned}$$
(2.5)

As in many papers, we introduce the following new variables:

$$\begin{aligned} \textstyle\begin{cases} z(x,\rho,\varrho,t)=u_{t}(x,t-\varrho \rho ), \\ y(x,\rho,\varrho,t)=v_{t}(x,t-\varrho \rho ),\end{cases}\displaystyle \end{aligned}$$

then we obtain

$$\begin{aligned} \textstyle\begin{cases} \varrho z_{t}(x,\rho,\varrho,t)+z_{\rho }(x,\rho,\varrho,t)=0, \\ z(x,0,\varrho,t)=u_{t}(x,t),\end{cases}\displaystyle \end{aligned}$$
(2.6)

and

$$\begin{aligned} \textstyle\begin{cases} \varrho y_{t}(x,\rho,\varrho,t)+y_{\rho }(x,\rho,\varrho,t)=0, \\ y(x,0,\varrho,t)=v_{t}(x,t).\end{cases}\displaystyle \end{aligned}$$
(2.7)

Consequently, problem (1.1) is equivalent to

$$\begin{aligned} \textstyle\begin{cases} u_{tt}-\Delta _{e}u+k_{1} u_{t}+\int _{\tau _{1}}^{ \tau _{2}}\mu _{1}(\varrho )z(x,1,\varrho,t )\,d\varrho =f_{1} ( u,v ), \\ v_{tt}-\Delta _{e}v+k_{2} v_{t}+\int _{\tau _{1}}^{\tau _{2}}\mu _{2}( \varrho )y(x,1,\varrho,t )\,d\varrho =f_{2} ( u,v ), \\ \varrho z_{t}(x,\rho,\varrho,t)+z_{\rho }(x,\rho,\varrho,t)=0, \\ \varrho y_{t}(x,\rho,\varrho,t)+y_{\rho }(x,\rho,\varrho,t)=0, \end{cases}\displaystyle \end{aligned}$$
(2.8)

with the initial data and boundary conditions

$$\begin{aligned} \textstyle\begin{cases} (u(x,0),v(x,0))=(u_{0}(x),v_{0}(x))\quad \text{in }\Omega, \\ (u_{t}(x,0),v_{t}(x,0))=(u_{1}(x),v_{1}(x))\quad \text{in }\Omega, \\ (u_{t}(x,-t),v_{t}(x,-t))=(f_{0}(x,t),g_{0}(x,t)) \quad\text{in }\Omega \times (0,\tau _{2}), \\ u(x,t)=v(x,t)=0 \quad\text{in }\partial \Omega \times (0,\infty ), \\ ( z(x,\rho,\varrho,0),y(x,\rho,\varrho,0) ) =(f_{0}(x, \rho \varrho ),g_{0}(x,\rho \varrho ))\quad \text{in }\Omega \times ( 0,1 ) \times (0,\tau _{2}),\end{cases}\displaystyle \end{aligned}$$
(2.9)

where

$$\begin{aligned} (x,\rho,\varrho,t)\in \Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) \times ( 0,\infty ). \end{aligned}$$

The energy associated with problem (2.8) is defined by

$$\begin{aligned} \begin{aligned} E ( t ) ={}&\frac{1}{2} \bigl[ \Vert u_{t} \Vert ^{2}+ \Vert v_{t} \Vert ^{2}+\mu \bigl( \Vert \nabla u \Vert ^{2}+ \Vert \nabla v \Vert ^{2} \bigr) + ( \lambda +\mu ) \bigl( \Vert \operatorname{div} u \Vert ^{2}+ \Vert \operatorname{div} v \Vert ^{2} \bigr) \bigr] \\ &{}+\frac{1}{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,\rho,\varrho,t ) \bigr\vert ^{2}+ \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,\rho,\varrho,t ) \bigr\vert ^{2} \bigr] \,d \varrho\, d\rho \,dx \\ &{}- \int _{\Omega }F ( u,v ) \,dx. \end{aligned} \end{aligned}$$
(2.10)

First, we prove in the following theorem, the result of energy identity.

Lemma 2.1

Assume that

$$\begin{aligned} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{i} ( \varrho ) \bigr\vert \,d\varrho < k_{i},\quad i=1,2. \end{aligned}$$
(2.11)

Then the energy modified defined by (2.10) satisfies, along the solution \(( u,v,z,y ) \) of (2.8), the estimate

$$\begin{aligned} \begin{aligned} \frac{d}{dt}E ( t )&\leq - \biggl[ k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \Vert u_{t} \Vert ^{2}- \biggl[ k_{2}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \Vert v_{t} \Vert ^{2}\\ &\leq 0. \end{aligned} \end{aligned}$$
(2.12)

Proof

First multiplying equation (2.8)1 by \(u_{t}\) and integrating by parts over Ω, we obtain

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \Vert u_{t} \Vert ^{2}+\mu \Vert \nabla u \Vert ^{2}+ ( \lambda + \mu ) \Vert \operatorname{div} u \Vert ^{2} \bigr] +k_{1} \Vert u_{t} \Vert ^{2} \\ &\qquad{}+ \int _{\Omega } u_{t} \int _{\tau _{1}}^{ \tau _{2}}|\mu _{1}(\varrho ) z ( x,1, \varrho,t ) \,d \varrho \,dx \\ &\quad= \int _{\Omega }u_{t}f_{1} ( u,v )\,dx. \end{aligned} \end{aligned}$$
(2.13)

Similarly, multiplying equation (2.8)2 by \(v_{t}\) and integrating over Ω, we obtain

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \Vert v_{t} \Vert ^{2}+\mu \Vert \nabla v \Vert ^{2}+ ( \lambda + \mu ) \Vert \operatorname{div} v \Vert ^{2} \bigr] +k_{1} \Vert v_{t} \Vert ^{2} \\ &\qquad{}+ \int _{\Omega } v_{t} \int _{\tau _{1}}^{ \tau _{2}}|\mu _{2}(\varrho ) y ( x,1, \varrho,t ) \,d \varrho \,dx \\ &\quad= \int _{\Omega }v_{t}f_{2} ( u,v )\,dx. \end{aligned} \end{aligned}$$
(2.14)

Multiplying equation (2.8)3 by \(\vert \mu _{1} ( \varrho ) \vert z ( x, \rho,\varrho,t ) \) and integrating by parts over \(\Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) \), we obtain

$$\begin{aligned} \begin{aligned} & \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z ( x,\rho,\varrho,t ) z_{t} ( x,\rho,\varrho,t ) \,d\varrho \,d\rho \,dx \\ &\quad=- \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z ( x,\rho, \varrho,t ) z_{\rho } ( x,\rho,\varrho,t ) \,d \varrho \,d\rho \,dx, \end{aligned} \end{aligned}$$

therefore

$$\begin{aligned} \begin{aligned} &\frac{d}{dt}\frac{1}{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx \\ &\quad=-\frac{1}{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \frac{d}{d\rho } \bigl\vert z ( x,\rho, \varrho,t ) \bigr\vert ^{2}\,d \varrho \,d\rho \,dx \\ &\quad=-\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx+ \frac{1 }{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert u_{t} ( x,t ) \bigr\vert ^{2}\,d\varrho \,dx. \end{aligned} \end{aligned}$$
(2.15)

Similarly, multiplying the fourth equation of (2.8) by \(\vert \mu _{2} ( \varrho ) \vert y ( x, \rho,\varrho,t ) \) and integrating over \(\Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} )\), we obtain

$$\begin{aligned} \begin{aligned} &\frac{d}{dt} \biggl( \frac{1}{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx \biggr) \\ &\quad =-\frac{1}{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \frac{d}{d\rho } \bigl\vert y ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx \\ &\quad=-\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx+ \frac{1 }{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert v_{t} ( x,t ) \bigr\vert ^{2}\,d\varrho \,dx. \end{aligned} \end{aligned}$$
(2.16)

For the source term, we have

$$\begin{aligned} & \int _{\Omega }u_{t}f_{1} ( u,v ) \,dx+ \int _{ \Omega }v_{t}f_{2} ( u,v ) \,dx \\ &\quad = \int _{\Omega }u_{t} \bigl( \alpha \vert u+v \vert ^{p-1)}(u+v)+\beta \vert u \vert ^{\frac{p-3}{2}}u \vert v \vert ^{\frac{p+1}{2}} \bigr) \\ \begin{aligned} &\qquad{}+ \int _{\Omega }v_{t} \bigl( \alpha \vert u+v \vert ^{p-1}(u+v)+\beta \vert v \vert ^{ \frac{p-3}{2}}v \vert u \vert ^{\frac{p+1}{2}} \bigr) \\ &\quad= \int _{\Omega } \bigl( \alpha \vert u+v \vert ^{p-1}(u+v) ( u_{t}+v_{t} ) +\beta \bigl( \vert u \vert ^{\frac{p-3}{2}}u.u_{t} \bigr) \vert v \vert ^{ \frac{p+1}{2}} \end{aligned}\\ &\qquad{}+ \beta \bigl( \vert v \vert ^{\frac{p-3}{2}}v.v_{t} \bigr) \vert u \vert ^{ \frac{p+1}{2}} \bigr) \,dx \\ &\quad=\frac{d}{dt} \int _{\Omega } \biggl( \frac{\alpha }{p+1} \vert u+v \vert ^{p+1}+ \frac{2\beta }{p+1} \vert uv \vert ^{\frac{p+1}{2}} \biggr) \,dx=\frac{d}{dt} \int _{ \Omega }F ( u,v ) \,dx. \end{aligned}$$
(2.17)

By collecting the previous equations (2.13)–(2.17), we get

$$\begin{aligned} \begin{aligned} \frac{d}{dt}E ( t ) ={}&{-}k_{1} \Vert u_{t} \Vert ^{2}- \int _{\Omega } u_{t} \int _{\tau _{1}}^{\tau _{2}} \mu _{1} ( \varrho ) z ( x,1,\varrho,t ) \,d \varrho \,dx \\ &{}-k_{2} \Vert v_{t} \Vert ^{2}- \int _{\Omega } v_{t} \int _{ \tau _{1}}^{\tau _{2}}\mu _{2} ( \varrho ) y ( x,1, \varrho,t ) \,d\varrho \,dx \\ &{}-\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,1, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx+\frac{1 }{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \Vert u_{t} \Vert ^{2} \\ &{}-\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,1, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx+\frac{1 }{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \Vert v_{t} \Vert ^{2}. \end{aligned} \end{aligned}$$
(2.18)

Using Young’s inequality, we obtain

$$\begin{aligned} \begin{aligned} & \int _{\Omega } u_{t} \int _{\tau _{1}}^{\tau _{2}}\mu _{1} ( \varrho ) z ( x,1,\varrho,t ) \,d\varrho \,dx \\ &\quad\leq \frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \Vert u_{t} \Vert + \frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,1,\varrho,t ) \bigr\vert ^{2}\,d \varrho \,dx, \end{aligned} \end{aligned}$$
(2.19)

similarly

$$\begin{aligned} \begin{aligned} & \int _{\Omega } v_{t} \int _{\tau _{1}}^{\tau _{2}}\mu _{2} ( \varrho ) y ( x,1,\varrho,t ) \,d\varrho \,dx \\ &\quad\leq \frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \Vert v_{t} \Vert + \frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,1,\varrho,t ) \bigr\vert ^{2}\,d \varrho \,dx, \end{aligned} \end{aligned}$$
(2.20)

This completes the proof. □

Global existence

Theorem 3.1

(Global existence)

Let \((u_{0},v_{0})\in (H^{2}(\Omega ) \cap H_{0}^{1}(\Omega ))^{2}\), \((u_{1},v_{1})\in (H_{0}^{1}(\Omega ))^{2}\) and \((f_{0},g_{0})\in (H^{1}(\Omega \times (0,1)\times ( \tau _{1}, \tau _{2} ) ))^{2}\) satisfying the compatibility condition

$$\begin{aligned} \bigl(f_{0}(\cdot,0),g_{0}(\cdot,0) \bigr)=(u_{1},v_{1}). \end{aligned}$$

Assume that (A1)–(A2) hold. Then, problem (2.8)(2.9) admits a weak solution such that \(u,v\in L^{\infty }(0,\infty;H^{2}(\Omega ) \cap H_{0}^{1}(\Omega ))\), \(u_{t},v_{t}\in L^{\infty }(0,\infty; H_{0}^{1}(\Omega ))\), and \(u_{tt},v_{tt}\in L^{2}(0,\infty,H_{0}^{1}(\Omega ))\).

Throughout this section we assume \(( u_{0},v_{0} ) \in ( H^{2}(\Omega )\cap H_{0}^{1}( \Omega ) ) ^{2}\), \(( u_{1},v_{1} ) \in ( H_{0}^{1}(\Omega ) ) ^{2}\) and \(f_{0},g_{0}\in H^{2}(\Omega;H^{1} ( 0,1 ) )\cap H_{0}^{1}( \Omega;H^{1} ( 0,1 ) )\). We employ the Galerkin method to construct a global solution. Let \(T>0\) be fixed and denote by \(V_{k}\) the space generated by \(\{ w_{1},w_{2},\ldots,w_{k} \} \), where the set \(\{ w_{k},k\in \mathbb{N} \} \) is a basis of \(H^{2}(\Omega )\cap H_{0}^{1}(\Omega )\). Now, we define for \(1\leq j\leq k\) the sequence \(\phi _{j} ( x,\rho ) \) as follows:

$$\begin{aligned} \phi _{j} ( x,0 ) =w_{j}. \end{aligned}$$
(3.1)

Then, we may extend \(\phi _{j} ( x,0 ) \) by \(\phi _{j} ( x,\rho ) \) over \(L^{2} ( \Omega \times [ 0,1 ] ) \) and denote by \(Z_{k}\) the space generated by \(\{ \phi _{1},\phi _{2},\ldots,\phi _{k} \} \). We construct approximate solutions \((u^{k},v^{k},z^{k},y^{k} ) \) \(k=1,2,\ldots \) in the form

$$\begin{aligned} &u^{k} ( t ) =\sum_{j=1}^{k}g_{jk} ( t ) w_{j} ( x ), \qquad z^{k} ( t ) =\sum _{j=1}^{k}c_{jk} ( t ) \phi _{j} ( x,\rho,\varrho ) \end{aligned}$$
(3.2)
$$\begin{aligned} &v^{k} ( t ) =\sum_{j=1}^{k}h_{jk} ( t ) w_{j} ( x ), \qquad y^{k} ( t ) =\sum _{j=1}^{k}\,d_{jk} ( t ) \phi _{j} ( x,\rho,\varrho ), \end{aligned}$$
(3.3)

where \(g_{jk,}h_{jk},c_{jk}\), and \(d_{jk}\), \(j=1,2,\ldots\) , are determined by the following ordinary differential equations:

$$\begin{aligned} \textstyle\begin{cases} \langle u_{tt}^{k},w_{j} \rangle +\mu \langle \nabla u^{k},\nabla w_{j} \rangle + ( \lambda +\mu ) \langle \operatorname{div} u^{k},\operatorname{div} w_{j} \rangle +k_{1} \langle u_{t}^{k}, w_{j} \rangle \\ \quad{}+ \langle \int _{\tau _{1}}^{\tau _{2}} \vert \mu _{1} ( \varrho ) \vert z^{k} ( x,1,\varrho,t ) \,d\varrho, w_{j} \rangle = \langle f_{1} ( u^{k},v^{k} ),w_{j} \rangle,\quad 1\leq j\leq k, \\ \langle v_{tt}^{k},w_{j} \rangle +\mu \langle \nabla v^{k},\nabla w_{j} \rangle + ( \lambda +\mu ) \langle \operatorname{div} v^{k}, \operatorname{div} w_{j} \rangle +k_{2} \langle v_{t}^{k},w_{j} \rangle \\ \quad{}+ \langle \int _{\tau _{1}}^{\tau _{2}} \vert \mu _{2} ( \varrho ) \vert y^{k} ( x,1,\varrho,t ) \,d\varrho, w_{j} \rangle = \langle f_{2} ( u^{k},v^{k} ),w_{j} \rangle, \quad 1\leq j\leq k, \\ \langle \varrho z_{t}^{k}+\frac{\partial }{\partial \rho }z^{k}, \phi _{j} \rangle =0, \quad 1\leq j\leq k, \\ \langle \varrho y_{t}^{k}+\frac{\partial }{\partial \rho }y^{k}, \phi _{j} \rangle =0,\quad 1\leq j\leq k, \end{cases}\displaystyle \end{aligned}$$
(3.4)

and

$$\begin{aligned} \textstyle\begin{cases} ( u^{k} ( 0 ),u_{t,k} ( 0 ) ) = ( u_{0k},u_{1k} ), \\ ( v^{k} ( 0 ),v_{t,k} ( 0 ) ) = ( u_{0k},u_{1k} ), \\ z^{k} ( x,0,\varrho,t ) =u_{t}^{k} ( x,t ), \\ y^{k} ( x,0,\varrho,t ) =v_{t}^{k} ( x,t ). \end{cases}\displaystyle \end{aligned}$$
(3.5)

Suppose that

$$\begin{aligned} w_{j}\in H^{2} ( \Omega ). \end{aligned}$$
(3.6)

We choose \(u_{0}^{k}\), \(v_{0}^{k}\), \(u_{1}^{k}\) and \(v_{1}^{k}\in [ w_{1},\ldots,w_{k} ] \), where

$$\begin{aligned} &u^{k} ( 0 ) =u_{0}^{k}=\sum _{j=1}^{k} ( u_{0},w_{j} ) w_{j}\rightarrow u_{0}\quad \text{in }H^{2}(\Omega )\cap H_{0}^{1}( \Omega )\text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.7)
$$\begin{aligned} &u_{t}^{k} ( 0 ) =u_{1}^{k}= \sum_{j=1}^{k} ( u_{1},w_{j} ) w_{j}\rightarrow u_{1}\quad\text{in } H_{0}^{1}( \Omega ) \text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.8)
$$\begin{aligned} & v^{k} ( 0 ) =v_{0}^{k}=\sum _{j=1}^{k} ( v_{0},w_{j} ) w_{j}\rightarrow v_{0}\quad\text{in }H^{2}(\Omega )\cap H_{0}^{1}( \Omega )\text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.9)
$$\begin{aligned} & v_{t}^{k} ( 0 ) =v_{1}^{k}= \sum_{j=1}^{k} ( v_{1},w_{j} ) w_{j}\rightarrow v_{1}\quad \text{in } H_{0}^{1}( \Omega ) \text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.10)
$$\begin{aligned} &z^{k} ( \rho,0 ) =z_{0}^{k}=\sum _{j=1}^{k} ( f_{0}, \phi _{j} ) \phi _{j}\rightarrow f_{0}\quad\text{in }L^{2} \bigl( \Omega \times (0,1)\times (\tau _{1},\tau _{2}) \bigr) \text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.11)
$$\begin{aligned} & y^{k} ( \rho,0 ) =y_{0}^{k}=\sum _{j=1}^{k} ( g_{0}, \phi _{j} ) \phi _{j}\rightarrow g_{0}\quad\text{in }L^{2} \bigl( \Omega \times (0,1)\times (\tau _{1},\tau _{2}) \bigr) \text{ as } k\rightarrow +\infty. \end{aligned}$$
(3.12)

By virtue of the theory of ordinary differential equations, system (3.4)–(3.12) has a unique local solution which is extended to a maximal interval \([ 0,T_{k} [ \) (with \(0< T_{k}\leq +\infty \)). We can utilize a standard compactness argument for the limiting procedure.

The first estimate.

Lemma 3.2

There exists a constant \(T>0\) such that the approximate solution satisfies, for all \(k\geq 1\):

$$\begin{aligned} & u^{k},v^{k} \quad\textit{are bounded in } L^{\infty } \bigl( 0,T;H_{0}^{1} ( \Omega ) \bigr), \end{aligned}$$
(3.13)
$$\begin{aligned} & u_{t}^{k},v_{t}^{k}\quad \textit{are bounded in }L^{\infty } \bigl( 0,T;L^{2} (\Omega ) \bigr), \end{aligned}$$
(3.14)
$$\begin{aligned} & z^{k} ( x,\rho,\varrho,t ),y^{k} ( x,\rho, \varrho,t ) \quad\textit{are bounded in } L^{\infty } \bigl( 0,T; L^{2} \bigl(\Omega \times ( 0,1 ) \times ( \tau _{1}, \tau _{2} ) \bigr) \bigr). \end{aligned}$$
(3.15)

Proof

Multiplying the first and second equations of (3.4) by \(( g_{jk}^{\prime } ) \) and \(( h_{jk}^{\prime } ) \) respectively and summing with respect to j, we obtain

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u^{k} \bigr\Vert ^{2}+ ( \lambda + \mu ) \bigl\Vert \operatorname{div} u^{k} \bigr\Vert ^{2}+ \bigl\Vert v_{t}^{k} \bigr\Vert ^{2}\\ &\quad{}+\mu \bigl\Vert \nabla v^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v^{k} \bigr\Vert ^{2} \bigr] \\ &\quad{}+ \int _{\Omega } u_{t}^{k} \int _{\tau _{1}}^{\tau _{2}}\mu _{1} ( \varrho ) z^{k} ( x,1,\varrho,t ) \,d \varrho \,dx+ \int _{\Omega } v_{t}^{k} \int _{\tau _{1}}^{\tau _{2}}\mu _{2} ( \varrho ) y^{k} ( x,1,\varrho,t ) \,d \varrho \,dx \\ &\quad{}+k_{1} \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} = \bigl\langle f_{1} \bigl( u^{k},v^{k} \bigr),u_{t}^{k} \bigr\rangle + \bigl\langle f_{2} \bigl( u^{k},v^{k} \bigr),v_{t}^{k} \bigr\rangle ,\quad 1\leq j\leq k. \end{aligned} \end{aligned}$$
(3.16)

Multiplying (3.4) by \(\vert \mu _{1} ( \varrho ) \vert ( c_{jk} ) \) and \(\vert \mu _{2} ( \varrho ) \vert ( d_{jk} ) \) respectively, iterating over \(\Omega \times ( 0,1 ) \times ( \tau _{1,}\tau _{2} ) \), and summing with respect to j, we obtain

$$\begin{aligned} \begin{aligned} &\frac{1 }{2}\frac{d}{dt} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z^{k} \bigr\vert ^{2}\,d \varrho \,d\rho \,dx\\ &\qquad{}+\frac{1}{2} \frac{d}{dt} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y^{k} \bigr\vert ^{2}\,d \varrho \,d\rho \,dx \\ &\quad=-\frac{1 }{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z^{k} ( x,1, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx+ \frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{t}^{k} \bigr\Vert ^{2} \\ &\qquad{}-\frac{1}{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y^{k} ( x,1, \varrho,t ) \bigr\vert ^{2}\,d \varrho \,dx+\frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{t}^{k} \bigr\Vert ^{2}. \end{aligned} \end{aligned}$$
(3.17)

By summing (3.16)–(3.17), we have

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u^{k} \bigr\Vert ^{2}+ ( \lambda + \mu ) \Vert \operatorname{div} u_{k} \Vert ^{2}+ \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} +\mu \bigl\Vert \nabla v^{k} \bigr\Vert ^{2} ( \lambda +\mu ) \bigl\Vert \operatorname{div} v^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{}+\frac{1 }{2}\frac{d}{dt} \biggl[ \int _{\Omega } \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z^{k} ( x,\rho, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx \\ &\qquad{}+ \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y^{k} ( x,\rho, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx \biggr] \\ \begin{aligned} &\qquad{}+k_{1} \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} \\ &\qquad{}+\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z^{k} ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx-\frac{1 }{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{t}^{k} \bigr\Vert ^{2} \end{aligned} \\ &\qquad{}+\frac{1 }{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y^{k} (x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx- \frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} \\ &\quad= \int _{\Omega } \bigl( u_{t}^{k}f_{1} \bigl( u^{k},v^{k} \bigr) +v_{t}^{k}f_{2} \bigl( u^{k},v^{k} \bigr) \bigr) \,dx \\ &\qquad{}- \int _{\Omega } u_{t}^{k} \int _{\tau _{1}}^{\tau _{2}}\mu _{1} ( \varrho ) z_{k} ( x,1,\varrho,t ) \,d \varrho \,dx- \int _{\Omega } v_{t}^{k} \int _{\tau _{1}}^{\tau _{2}}\mu _{2} ( \varrho ) y_{k} ( x,1,\varrho,t ) \,d \varrho \,dx. \end{aligned}$$
(3.18)

By using Holder and Young’s inequalities, we have

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u^{k} \bigr\Vert ^{2}+ ( \lambda + \mu ) \bigl\Vert \operatorname{div} u^{k} \bigr\Vert ^{2}\\ &\qquad{}+ \bigl\Vert v_{t}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla v^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{}+\frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z^{k} ( x,\rho, \varrho,t ) \bigr\Vert ^{2} \\ &\qquad{}+\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y^{k} ( x,\rho,\varrho,t ) \bigr\Vert ^{2} \bigr]\,d\varrho \,d\rho \\ &\qquad{}+ \biggl[ k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+ \biggl[ k_{2}- \biggl( \int _{ \tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} \\ &\quad\leq \int _{\Omega } \bigl( u_{t}^{k}f_{1} \bigl( u^{k},v^{k} \bigr) +v_{t}^{k}f_{2} \bigl( u^{k},v^{k} \bigr) \bigr) \,dx. \end{aligned} \end{aligned}$$
(3.19)

Integrating over \((0,t),0< t< T_{k}\), we obtain

$$\begin{aligned} \begin{aligned} & \bigl\Vert u_{t}^{k} ( t ) \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u^{k} ( t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u^{k} ( t ) \bigr\Vert ^{2}+ \bigl\Vert v_{t}^{k} ( t ) \bigr\Vert ^{2}\\ &\qquad{}+ \mu \bigl\Vert \nabla v^{k} ( t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+ \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z^{k} ( x,\rho, \varrho,t ) \bigr\Vert ^{2} +\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y^{k} ( x,\rho,\varrho,t ) \bigr\Vert ^{2} \bigr]\,d\varrho \,d \rho \\ &\qquad{}+2 \biggl[ k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \int _{0}^{t} \bigl\Vert u_{t}^{k} ( s ) \bigr\Vert ^{2}\,ds\\ &\qquad{}+2 \biggl[ k_{2}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \int _{0}^{t} \bigl\Vert v_{t}^{k} ( s ) \bigr\Vert ^{2}\,ds \\ &\quad\leq C_{0}+c \int _{0}^{t} \int _{\Omega } \bigl[ u_{t}^{k}f_{1} \bigl( u^{k},v^{k} \bigr)+v_{t}^{k}f_{2} \bigl( u^{k},v^{k} \bigr) \bigr]\,dx\,ds, \end{aligned} \end{aligned}$$
(3.20)

where

$$\begin{aligned} C_{0}={}&C \bigl( \bigl\Vert u_{0}^{k} \bigr\Vert _{H^{1} ( \Omega ) }, \bigl\Vert v_{0}^{k} \bigr\Vert _{H^{1} ( \Omega )}, \bigl\Vert u_{1}^{k} \bigr\Vert _{L^{2} ( \Omega ) }, \bigl\Vert v_{1}^{k} \bigr\Vert _{L^{2} ( \Omega ) },\\ &{} \Vert f_{0} \Vert _{L^{2} (\Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) ) }, \Vert g_{0} \Vert _{L^{2} (\Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) ) } \bigr) \end{aligned}$$

is a positive constant. We just need to estimate the right-hand terms of (3.20). Applying Hölder’s inequality, Sobolev’s embedding theorem, and Young’s inequality, we obtain

$$\begin{aligned} \begin{aligned} & \biggl\vert \int _{0}^{t} \int _{\Omega }u_{t}^{k} ( s ) f_{1} \bigl( u^{k} ( s ),v^{k} ( s ) \bigr) \,ds \biggr\vert \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\vert u^{k} ( s ) \bigr\vert ^{p}+ \bigl\vert v^{k} ( s ) \bigr\vert ^{p}+ \bigl\vert u^{k} ( s ) \bigr\vert ^{ \frac{p-1}{2}} \bigl\vert v^{k} ( s ) \bigr\vert ^{ \frac{p+1}{2}} \bigr) \bigl\vert u_{t}^{k} ( s ) \bigr\vert \,ds \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\Vert u^{k} ( s ) \bigr\Vert _{2p}^{p}+ \bigl\Vert v^{k} ( s ) \bigr\Vert _{2p}^{p}\\ &\qquad{}+ \bigl\Vert u^{k} ( s ) \bigr\Vert _{3 ( p-1 ) }^{\frac{p-1}{2}} \bigl\Vert v^{k} ( s ) \bigr\Vert _{\frac{3 ( p+1 ) }{2}}^{\frac{p+1}{2}} \bigr) \bigl\Vert u_{t}^{k} ( s ) ( s ) \bigr\Vert _{2}\,ds \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{p}+ \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{p}\\ &\qquad{}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{\frac{p-1}{2}} \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{\frac{p+1}{2}} \bigr) \bigl\Vert u_{t}^{k} ( s ) ( s ) \bigr\Vert _{2}\,ds \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\Vert u_{t}^{k} ( s ) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{2p}+ \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{2p}\\ &\qquad{}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{ ( p-1 ) } \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{ ( p+1 ) } \bigr) \,ds, \end{aligned} \end{aligned}$$
(3.21)

when we have used in (3.21) the Sobolev imbedding in (2.5) and the fact when \(n=3\) then \(2p=3 ( p-1 ) =\frac{3 ( p+1 ) }{2}=6\).

Likewise, we obtain

$$\begin{aligned} \begin{aligned} & \biggl\vert \int _{0}^{t} \int _{\Omega }v_{t}^{k} ( s ) f_{2} \bigl( u^{k} ( s ),v^{k} ( s ) \bigr) \,ds \biggr\vert \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\Vert v_{t}^{k} ( s ) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{2p}+ \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{2p}\\ &\qquad{}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{p+1} \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{p-1} \bigr) \,ds. \end{aligned} \end{aligned}$$
(3.22)

Let

$$\begin{aligned} X_{k} ( t ) = \bigl\Vert u_{t}^{k} ( t ) \bigr\Vert _{2}^{2}+ \bigl\Vert v_{t}^{k} ( t ) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u^{k} ( t ) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla v^{k} ( t ) \bigr\Vert _{2}^{2}. \end{aligned}$$
(3.23)

From assumptions of Lemma 2.1, we can find positive constants such that

$$\begin{aligned} \begin{aligned} &X_{k} ( t ) +c_{1} \bigl\Vert \operatorname{div} u^{k} ( t ) \bigr\Vert ^{2}+c_{1} \bigl\Vert \operatorname{div} v^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+c_{2} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z^{k} ( x,\rho, \varrho,t ) \bigr\Vert ^{2} + \varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y^{k} ( x,\rho, \varrho,t ) \bigr\Vert ^{2} \bigr]\,d\varrho \,d\rho \\ &\qquad{}+c_{3} \int _{0}^{t} \bigl\Vert u_{t}^{k} ( s ) \bigr\Vert ^{2}\,ds+c_{3} \int _{0}^{t} \bigl\Vert v_{t}^{k} ( s ) \bigr\Vert ^{2}\,ds \\ &\quad\leq C_{0}+c \int _{0}^{t} \int _{\Omega } \bigl( X_{k} ( t ) \bigr) ^{p} \,dx\,ds. \end{aligned} \end{aligned}$$
(3.24)

Particulary, we have

$$\begin{aligned} X_{k} ( t ) \leq C_{0}+c \int _{0}^{t} \int _{\Omega } \bigl( X_{k} ( t ) \bigr) ^{p} \,dx\,ds. \end{aligned}$$
(3.25)

Using Gronwall-type inequality, we can get

$$\begin{aligned} X_{k} ( t ) \leq \bigl[ C_{0}- ( p-1 ) Ct \bigr]^{-1/ ( p-1 ) }. \end{aligned}$$
(3.26)

Thus, we deduce from (3.26) that there exists a time \(T>0\) such that

$$\begin{aligned} X_{k} ( t ) \leq C_{1},\quad \forall t\in [ 0,T ], \end{aligned}$$
(3.27)

where \(C_{1}\) is a positive constant independent of k. Then inequality (3.27) established the first two parts of lemma. The last part of lemma immediately follows from (3.24). □

The second estimate: First, we are going to estimate \(u_{tt}^{k} ( 0 ) \) and \(v_{tt}^{k} ( 0 ) \). Testing the first and second equations in (3.4) by \(g_{j,k}^{\prime \prime } ( t ) \) and \(h_{j,k}^{\prime \prime } ( t ) \) respectively and taking \(t=0\), we obtain

$$\begin{aligned} \begin{aligned}& \bigl\Vert u_{tt}^{k} ( 0 ) \bigr\Vert _{2}^{2}+ \bigl\Vert v_{tt}^{k} ( 0 ) \bigr\Vert _{2}^{2} \\ &\quad\leq \bigl\Vert \Delta u_{0}^{k} \bigr\Vert _{2}^{2}+ \bigl\Vert \Delta v_{0}^{k} \bigr\Vert _{2}^{2}+c \bigl\Vert \operatorname{div} u_{0}^{k} \bigr\Vert _{2}+c \bigl\Vert \operatorname{div} v_{0}^{k} \bigr\Vert _{2}^{2}+ \bigl\Vert \Delta u_{t}^{k} ( 0 ) \bigr\Vert _{2}^{2} \\ &\qquad{}+ \bigl\Vert \Delta v_{t}^{k} ( 0 ) \bigr\Vert _{2}^{2}+c \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert \Delta z^{k} (x,1, \varrho,0 ) \bigr\vert ^{2} \,d\varrho \,dx \\ &\qquad{}+c \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert \Delta y^{k} ( x,1, \varrho,0 ) \bigr\vert ^{2} \,d\varrho \,dx. \end{aligned} \end{aligned}$$
(3.28)

From (3.7)–(3.12), we have

$$\begin{aligned} \bigl\Vert u_{tt}^{k} ( 0 ) \bigr\Vert _{2}^{2}+ \bigl\Vert v_{tt}^{k} ( 0 ) \bigr\Vert _{2}^{2}\leq C. \end{aligned}$$
(3.29)

In order to calculate the second estimate, we take the derivatives of the first and second equations of system (3.4) with respect to t, we get

$$\begin{aligned} \begin{aligned} & \bigl\langle u_{ttt}^{k},w_{j} \bigr\rangle +\mu \bigl\langle \nabla u_{t}^{k},\nabla w_{j} \bigr\rangle + ( \lambda +\mu ) \bigl\langle \operatorname{div} u_{t}^{k}, \operatorname{div} w_{j} \bigr\rangle \\ &\quad{}+k_{1} \bigl\langle u_{tt}^{k}, w_{j} \bigr\rangle + \biggl\langle \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z_{t}^{k} ( x,1,\varrho,t ) \,d \varrho, w_{j} \biggr\rangle = \bigl\langle Df_{1} ( u_{k},v_{k} ),w_{j} \bigr\rangle ,\quad 1\leq j\leq k \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} & \bigl\langle v_{ttt}^{k},w_{j} \bigr\rangle +\mu \bigl\langle \nabla v_{t}^{k},\nabla w_{j} \bigr\rangle + ( \lambda +\mu ) \bigl\langle \operatorname{div} v_{t}^{k},\operatorname{div} w_{j} \bigr\rangle \\ &\quad{}+k_{2} \bigl\langle v_{tt}^{k}, w_{j} \bigr\rangle + \biggl\langle \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert y_{t}^{k} ( x,1,\varrho,t ) \,d \varrho, w_{j} \biggr\rangle = \bigl\langle Df_{2} ( u_{k},v_{k} ),w_{j} \bigr\rangle ,\quad 1\leq j\leq k. \end{aligned} \end{aligned}$$

Multiplying by \(( g_{j,k}^{\prime \prime } ( t ) ) \) and \(( h_{j,k}^{\prime \prime } ( t ) ) \) respectively and summing with respect to j from 1 to k, we obtain

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+\mu \bigl\Vert \nabla u_{t}^{k} ( t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} ( t ) \bigr\Vert ^{2} \bigr] +k_{1} \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+ \int _{\Omega } u_{tt}^{k} ( t ) \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z_{t}^{k} ( x,1,\varrho,t ) \,d\varrho \,dx\\ &\quad= \int _{\Omega }Df_{1} ( u_{k},v_{k} ) u_{tt}^{k} ( t ) \,dx,\quad 1 \leq j\leq k, \end{aligned} \end{aligned}$$
(3.30)

and

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} \bigr\Vert ^{2} \bigr] +k_{2} \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2} \\ &\qquad{} + \int _{\Omega } v_{tt}^{k} ( t ) \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert y_{t}^{k} ( x,1,\varrho,t ) \,d\varrho \,dx\\ &\quad = \int _{\Omega }Df_{2} ( u_{k},v_{k} ) v_{tt}^{k} ( t ) \,dx, \quad 1 \leq j\leq k. \end{aligned} \end{aligned}$$
(3.31)

Differentiating the third and fourth equations in (3.4) with respect to t, we get

$$\begin{aligned} &\biggl\langle \varrho z_{tt}^{k}+\frac{\partial }{\partial \rho } z_{t}^{k}, \phi _{j} \biggr\rangle =0,\quad 1\leq j \leq k, \\ &\biggl\langle \varrho y_{tt}^{k}+\frac{\partial }{\partial \rho } y_{t}^{k}, \phi _{j} \biggr\rangle =0, \quad 1 \leq j\leq k. \end{aligned}$$

Multiplying by \(\vert \mu _{1} ( \varrho ) \vert c_{jk}^{\prime }\) and \(\vert \mu _{2} ( \varrho ) \vert \,d_{jk}^{\prime }\) respectively, integrating over \(( 0,1 ) \times ( \tau _{1},\tau _{2} ) \), and summing over j from 1 to k, it follows that

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\rho, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx\\ &\quad{}+\frac{1}{2}\int _{0}^{1} \int _{\Omega } \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \frac{d}{d\rho } \bigl\vert z_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d \varrho \,d\rho \,dx=0, \\ & \frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx\\ &\quad{}+\frac{1}{2}\int _{0}^{1} \int _{\Omega } \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \frac{d}{d\rho } \bigl\vert y_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d \varrho \,d\rho \,dx=0, \end{aligned} \end{aligned}$$

then we obtain

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx\\ &\quad{}+ \frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx \\ &\quad{}-\frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}\,dx=0 \end{aligned} \end{aligned}$$
(3.32)

and

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx\\ &\quad{}+\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho\, dx \\ &\quad{} -\frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2}\,dx=0. \end{aligned} \end{aligned}$$
(3.33)

Taking the sum of (3.30), (3.31),(3.32), and (3.33), we get

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{} +\frac{1}{2}\frac{d}{dt} \biggl[ \int _{0}^{1} \int _{\Omega } \int _{ \tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \\ &\qquad{}+ \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \biggr] \\ &\qquad{} +k_{1} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho\, dx \\ &\qquad{} +\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,1, \varrho,t ) \bigr\vert ^{2}\,d\varrho\, dx \\ &\quad=\frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+\frac{1}{2} \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}- \int _{\Omega } u_{tt}^{k} ( t ) \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z_{t}^{k} ( x,1,\varrho,t ) \,d\varrho \, dx \\ &\qquad{}- \int _{\Omega } v_{tt}^{k} ( t ) \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert y_{t}^{k} ( x,1,\varrho,t ) \,d\varrho \, dx \\ &\qquad{}+ \int _{\Omega }Df_{1} ( u_{k},v_{k} ) u_{tt}^{k} ( t ) \,dx+ \int _{\Omega }Df_{2} ( u_{k},v_{k} ) v_{tt}^{k} ( t ) \,dx. \end{aligned}$$
(3.34)

Using Cauchy–Schwarz and Young’s inequalities, we conclude

$$\begin{aligned} \begin{aligned}& \biggl\vert \int _{\Omega } u_{tt}^{k} ( t ) \int _{\tau _{1}}^{\tau _{2}} \biggr\vert \mu _{1} ( \varrho ) \bigl\vert z_{t}^{k} ( x,1,\varrho,t ) \,d \varrho\, dx \bigr\vert \\ &\quad\leq \frac{1}{2} \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d \varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z_{t}^{k} ( x,1, \varrho,t ) \bigr\Vert ^{2}\,d\varrho. \end{aligned} \end{aligned}$$
(3.35)

Similarly,

$$\begin{aligned} \begin{aligned} &\biggl\vert \int _{\Omega } v_{tt}^{k} ( t ) \int _{\tau _{1}}^{\tau _{2}} \biggr\vert \mu _{2} ( \varrho ) \bigl\vert y_{t}^{k} ( x,1,\varrho,t ) \,d \varrho\, dx \bigr\vert \\ &\quad\leq \frac{1}{2} \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d \varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y_{t}^{k} ( x,1, \varrho,t ) \bigr\Vert ^{2}\,d\varrho. \end{aligned} \end{aligned}$$
(3.36)

For the source term

$$\begin{aligned} &\int _{\Omega }Df_{1} ( u_{k},v_{k} ) u_{tt}^{k} ( t ) \,dx \\ \begin{aligned} &\quad\leq C \bigl[ \bigl( \bigl\Vert u^{k} \bigr\Vert ^{p-1}+ \bigl\Vert v^{k} \bigr\Vert ^{p-1} \bigr) \bigl\Vert u_{t}^{k} \bigr\Vert + \bigl( \bigl\Vert u^{k} \bigr\Vert ^{p-1}+ \bigl\Vert v^{k} \bigr\Vert ^{p-1} \bigr) \bigl\Vert v_{t}^{k} \bigr\Vert \bigr] \bigl\Vert u_{tt}^{k} \bigr\Vert \\ &\quad\leq C \bigl[ \bigl\Vert u^{k} \bigr\Vert _{2} ^{2(p-1)}+ \bigl\Vert v^{k} \bigr\Vert _{2}^{2(p-1)}+ \bigl\Vert u_{t}^{k} \bigr\Vert _{2}^{2} + \bigl\Vert v_{t}^{k} \bigr\Vert _{2}^{2} \bigr] \bigl\Vert u_{tt}^{k} \bigr\Vert _{2} \end{aligned} \\ &\quad\leq \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert _{2}^{2}+c \end{aligned}$$
(3.37)

and

$$\begin{aligned} \begin{aligned} &\int _{\Omega }Df_{2} ( u_{k},v_{k} ) v_{tt}^{k} ( t ) \,dx\\ &\quad\leq C \bigl[ \bigl( \bigl\Vert u^{k} \bigr\Vert ^{p-1}+ \bigl\Vert v^{k} \bigr\Vert ^{p-1} \bigr) \bigl\Vert u_{t}^{k} \bigr\Vert + \bigl( \bigl\Vert u^{k} \bigr\Vert ^{p-1}+ \bigl\Vert v^{k} \bigr\Vert ^{p-1} \bigr) \bigl\Vert v_{t}^{k} \bigr\Vert \bigr] \bigl\Vert v_{tt}^{k} \bigr\Vert \\ &\quad\leq C \bigl[ \bigl\Vert u^{k} \bigr\Vert _{2} ^{2(p-1)}+ \bigl\Vert v^{k} \bigr\Vert _{2}^{2(p-1)}+ \bigl\Vert u_{t}^{k} \bigr\Vert _{2}^{2} + \bigl\Vert v_{t}^{k} \bigr\Vert _{2}^{2} \bigr] \bigl\Vert v_{tt}^{k} \bigr\Vert _{2} \\ &\quad\leq \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert _{2}^{2}+c. \end{aligned} \end{aligned}$$
(3.38)

Taking into account (3.35)–(3.38) into (3.34), we get

$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{} +\frac{1}{2}\frac{d}{dt} \biggl[ \int _{0}^{1} \int _{\Omega } \int _{ \tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\varrho,\rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx\\ &\qquad{}+ \frac{1}{2} \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\varrho, \rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \biggr] \\ &\qquad{}+k_{1} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z_{t}^{k} ( x,1,\varrho,t ) \bigr\Vert ^{2}\,d\varrho\\ &\qquad{} +\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y_{t}^{k} ( x,1,\varrho,t ) \bigr\Vert ^{2}\,d\varrho \\ &\quad \leq \frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+\frac{1}{2} \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{} + \biggl( \frac{\int _{\tau _{1}}^{\tau _{2}} \vert \mu _{1} ( \varrho ) \vert \,d\varrho }{2} \biggr) \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+ \biggl( \frac{\int _{\tau _{1}}^{\tau _{2}} \vert \mu _{2} ( \varrho ) \vert \,d\varrho }{2} \biggr) \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2} \\ & \qquad{}+\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z_{t}^{k} ( x,1, \varrho,t ) \bigr\Vert ^{2}\,d\varrho +\frac{1}{2} \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y_{t}^{k} ( x,1,\varrho,t ) \bigr\Vert ^{2}\,d \varrho \\ &\qquad{}+ \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2} + \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2}+c. \end{aligned} \end{aligned}$$

After simplification, we obtain

$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{} +\frac{1}{2}\frac{d}{dt} \biggl[ \int _{0}^{1} \int _{\Omega } \int _{ \tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\varrho,\rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \\ \begin{aligned} &\qquad{}+ \frac{1}{2} \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\varrho, \rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \biggr] \\ &\qquad{}+k_{1} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2} \end{aligned} \\ &\quad \leq \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+ \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+ \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2} + \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2}+c. \end{aligned}$$
(3.39)

Integrating (3.39) over \(( 0,t ) \), we get

$$\begin{aligned} & \bigl[ \bigl\Vert u_{tt}^{k} (t ) \bigr\Vert ^{2}+\mu \bigl\Vert \nabla u_{t}^{k} (t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} (t ) \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} (t ) \bigr\Vert ^{2}\\ &\qquad{}+\mu \bigl\Vert \nabla v_{t}^{k} (t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} (t ) \bigr\Vert ^{2} \bigr] \\ & \qquad{}\times\biggl[ \int _{0}^{1} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\varrho,\rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx\\ &\qquad{}+ \frac{1}{2} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \int _{\Omega } \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\varrho,\rho,t ) \bigr\vert ^{2}\,d \varrho \,d\rho\, dx \biggr] \\ &\qquad{}+ \biggl( k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr) \int _{0}^{t} \bigl\Vert u_{tt}^{k} (s ) \bigr\Vert ^{2}\,ds\\ &\qquad{}+ \biggl( k_{2}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr) \int _{0}^{t} \bigl\Vert v_{tt}^{k} (s ) \bigr\Vert ^{2}\,ds \\ &\quad\leq \bigl[ \bigl\Vert u_{tt}^{k} (0 ) \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u_{t}^{k} (0 ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} (0 ) \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} (0 ) \bigr\Vert ^{2}\\ &\qquad{}+\mu \bigl\Vert \nabla v_{t}^{k} (0 ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} (0 ) \bigr\Vert ^{2} \bigr] \\ &\qquad{}\times \biggl[ \int _{0}^{1} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\varrho,\rho,0 ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx\\ &\qquad{}+ \frac{1}{2} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \int _{\Omega } \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\varrho,\rho,0 ) \bigr\vert ^{2}\,d \varrho \,d\rho\, dx \biggr] \\ &\qquad{}+ \int _{0}^{t} \bigl\Vert u_{tt}^{k} ( s ) \bigr\Vert ^{2}\,ds + \int _{0}^{t} \bigl\Vert v_{tt}^{k} (s ) \bigr\Vert ^{2}\,ds+c. \end{aligned}$$

Taking

$$\begin{aligned} y^{k} ( t ) = \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \end{aligned}$$

and by using Gronwall’s inequality, we conclude that

$$\begin{aligned} &u_{t}^{k}\text{ and } v_{t}^{k} \text{ are bounded in }L^{ \infty } \bigl( 0,T;H_{0}^{1} ( \Omega ) \bigr), \end{aligned}$$
(3.40)
$$\begin{aligned} & u_{tt}^{k}\text{ and }v_{tt}^{k} \text{ are bounded in } L^{ \infty } \bigl( 0,T;L^{2} ( \Omega ) \bigr), \end{aligned}$$
(3.41)
$$\begin{aligned} &z_{t}^{k}\text{ and }y_{t}^{k} \text{ are bounded in } L^{ \infty } \bigl( 0,T;L^{2} \bigl( \Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) \bigr) \bigr). \end{aligned}$$
(3.42)

Applying Dunford–Pettis’ theorem, we deduce from (3.13), (3.14), (3.15), (3.40), (3.41), and (3.42), replacing the sequence \(u_{k}\) with a subsequence if necessary, that

$$\begin{aligned} &u^{k}\rightarrow u, v^{k}\rightarrow v \text{ weak-star in } L^{ \infty } \bigl( 0,T;H_{0}^{1} ( \Omega ) \bigr), \end{aligned}$$
(3.43)
$$\begin{aligned} &z^{k}\rightarrow z, y^{k}\rightarrow y\text{ weak-star in } L^{ \infty } \bigl( 0,T; L^{2} \bigl(\Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) \bigr) \bigr), \end{aligned}$$
(3.44)
$$\begin{aligned} &u_{t}^{k}\rightarrow u_{t}, v_{t}^{k}\rightarrow v_{t} \text{ weak-star in }L^{\infty } \bigl( 0,T;H_{0}^{1} ( \Omega ) \bigr), \end{aligned}$$
(3.45)
$$\begin{aligned} & u_{tt}^{k}\rightarrow u_{tt}, v_{tt}^{k}\rightarrow v_{tt} \text{ weak-star in } L^{\infty } \bigl( 0,T;L^{2} ( \Omega ) \bigr), \end{aligned}$$
(3.46)
$$\begin{aligned} &z_{t}^{k}\rightarrow z_{t}, y_{t}^{k}\rightarrow y_{t} \text{ weak-star in } L^{\infty } \bigl( 0,T;L^{2} \bigl( \Omega \times ( 0,1 )\times ( \tau _{1},\tau _{2} ) \bigr) \bigr). \end{aligned}$$
(3.47)

Corollary 3.3

The sequences of approximate solutions \(\lbrace u_{k},u_{k} \rbrace \) satisfy, as \(k\rightarrow \infty \),

$$\begin{aligned} \textstyle\begin{cases} f_{1} ( u_{k},v_{k} )\rightarrow f_{1} ( u,v )\textit{ strongly in } L^{\infty } ( 0,T;L^{2} ( \Omega ) ), \\ f_{2} ( u_{k},v_{k} )\rightarrow f_{2} ( u,v ) \textit{ strongly in } L^{\infty } ( 0,T;L^{2} ( \Omega ) ). \end{cases}\displaystyle \end{aligned}$$
(3.48)

Proof

The proof is similar to that of [11]. □

We can complete the proof of theorem as in [2].

General decay

In this section we prove that the solution of problem (2.8)–(2.9) decays generally to a trivial solution using the energy method and a suitable Lyapunov functional.

In the following, we present our main stability result.

Theorem 4.1

The solution of (2.8) satisfies, for two positive constants \(\alpha, \beta \), the estimate

$$\begin{aligned} E(t)\leq \beta e^{-\alpha t}, \quad\forall t \geq 0. \end{aligned}$$
(4.1)

To prove the desired result, we create a Lyapunov functional equivalent to E. For this, we define some functions that allow us to construct this Lyapunov function.

Lemma 4.2

Let \((u,v,z,y)\) be a solution of problem (2.8). Then the functional

$$\begin{aligned} \varphi ( t ) = \int _{\Omega }u ( t ) u_{t} ( t )\, dx+ \int _{\Omega }v ( t ) v_{t} ( t ) \,dx \end{aligned}$$
(4.2)

satisfies the estimate

$$\begin{aligned} \begin{aligned} \varphi ^{\prime } ( t ) \leq{}& \biggl(1+ \frac{C k_{1}^{2}}{\mu } \biggr) \bigl\Vert u_{t} ( t ) \bigr\Vert ^{2}+ \biggl(1+\frac{Ck_{2}}{\mu } \biggr) \bigl\Vert v_{t} ( t ) \bigr\Vert ^{2}-\frac{\mu }{2} \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}-\frac{\mu }{2} \bigl\Vert \nabla v ( t ) \bigr\Vert ^{2} \\ &{}- ( \lambda +\mu ) \bigl\Vert \operatorname{div} u ( t ) \bigr\Vert ^{2}- ( \lambda +\mu ) \bigl\Vert \operatorname{div} v ( t ) \bigr\Vert ^{2} \\ &{}+\frac{C k_{1}}{\mu } \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert \nabla z ( x,1,\varrho,t ) \bigr\vert ^{2}\,d \varrho\, dx\\ &{}+\frac{Ck_{2}}{\mu } \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert \nabla y ( x,1,\varrho,t ) \bigr\vert ^{2}\,d \varrho\, dx \\ &{}+ ( p+1 ) \int _{\Omega }F \bigl( u ( t ),v ( t ) \bigr) \,dx. \end{aligned} \end{aligned}$$
(4.3)

Proof

Taking the derivative of (4.2), we obtain

$$\begin{aligned} \varphi ^{\prime } ( t ) = \int _{\Omega } \bigl\vert u_{t} ( t ) \bigr\vert ^{2}\,dx+ \int _{\Omega }u ( t ) u_{tt} ( t ) \,dx+ \int _{\Omega } \bigl\vert v_{t} ( t ) \bigr\vert ^{2}\,dx+ \int _{\Omega }v ( t ) v_{tt} ( t )\,dx. \end{aligned}$$

From problem (2.8) and using integration by parts, we get

$$\begin{aligned} \begin{aligned} \varphi ^{\prime } ( t ) = {}&\bigl\Vert u_{t} ( t ) \bigr\Vert ^{2}+ \bigl\Vert v_{t} ( t ) \bigr\Vert ^{2} \\ &{}+ \int _{\Omega }u ( t ) \biggl( \Delta _{e}u-k_{1} u_{t}- \int _{\tau _{1}}^{\tau _{2}}\mu _{1}(\varrho ) z(x,1, \varrho,t)\,d \varrho +f_{1} ( u,v ) \biggr) \,dx \\ &{}+ \int _{\Omega }v ( t ) \biggl( \Delta _{e}v-k_{2} \Delta v_{t}- \int _{\tau _{1}}^{\tau _{2}}\mu _{2}(\varrho )\Delta y(x,1,\varrho,t)\,d \varrho +f_{2} ( u,v ) \biggr) \,dx \\ ={}& \bigl\Vert u_{t} ( t ) \bigr\Vert ^{2}+ \bigl\Vert v_{t} ( t ) \bigr\Vert ^{2}-k_{1} \int _{ \Omega } u u_{t}\,dx-k_{2} \int _{\Omega } v v_{t}\,dx \\ &{}-\mu \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}- ( \lambda +\mu ) \bigl\Vert \operatorname{div} u ( t ) \bigr\Vert ^{2}-\mu \bigl\Vert \nabla v ( t ) \bigr\Vert ^{2}- ( \lambda +\mu ) \bigl\Vert \operatorname{div} v ( t ) \bigr\Vert ^{2} \\ &{}- \int _{\Omega } u ( t ) \int _{\tau _{1}}^{\tau _{2}} \mu _{1}(\varrho ) z(x,1, \varrho,t)\,d\varrho - \int _{\Omega } v ( t ) \int _{\tau _{1}}^{\tau _{2}}\mu _{2}(\varrho ) y(x,1, \varrho,t)\,d\varrho \\ &{}+ \int _{\Omega }v ( t ) f_{2} ( u,v ) \,dx+ \int _{ \Omega }u ( t ) f_{1} ( u,v ) \,dx. \end{aligned} \end{aligned}$$
(4.4)

By using Hölder’s and Young’s inequalities, we have

$$\begin{aligned} \begin{aligned} & \int _{\Omega } u ( t ) \int _{\tau _{1}}^{ \tau _{2}}\mu _{1}(\varrho ) z(x,1, \varrho,t)\,d\varrho\, dx\\ &\quad \leq \frac{\varepsilon }{2} \bigl\Vert u ( t ) \bigr\Vert ^{2}+ \frac{1}{2\varepsilon } \int _{\Omega } \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1}( \varrho ) \bigr\vert z(x,1,\varrho,t)\,d\varrho \biggr) ^{2}\,dx \\ &\quad\leq \frac{C\varepsilon }{2} \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}+\frac{1}{2\varepsilon } \biggl( \int _{\tau _{1}}^{ \tau _{2}}{ \bigl\vert \mu _{1}( \varrho ) \bigr\vert }\,d\varrho \biggr) \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1}( \varrho ) \bigr\vert \bigl\vert \nabla z(x,1,\varrho,t) \bigr\vert ^{2}\,d \varrho \\ &\quad\leq \frac{C\varepsilon }{2} \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}+\frac{k_{1}}{2\varepsilon } \int _{\Omega } \int _{ \tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1}( \varrho ) \bigr\vert \bigl\vert z(x,1,\varrho,t) \bigr\vert ^{2} \,d\varrho. \end{aligned} \end{aligned}$$
(4.5)

Similarly,

$$\begin{aligned} \begin{aligned} &\int _{\Omega } v ( t ) \int _{\tau _{1}}^{ \tau _{2}}\mu _{1}(\varrho ) y(x,1, \varrho,t)\,d\varrho\, dx\\ &\quad\leq \frac{C\varepsilon }{2} \bigl\Vert \nabla v ( t ) \bigr\Vert ^{2}+\frac{k_{2}}{2\varepsilon } \int _{\Omega } \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1}( \varrho ) \bigr\vert \bigl\vert y(x,1,\varrho,t) \bigr\vert ^{2} \,d\varrho. \end{aligned} \end{aligned}$$
(4.6)

Young’s and Poincaré’s inequalities give

$$\begin{aligned} k_{1} \int _{\Omega } u ( t ) u_{t} ( t )\,dx\leq \frac{C\varepsilon }{2} \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}+ \frac{k_{1}^{2}}{2\varepsilon } \bigl\Vert u_{t} ( t ) \bigr\Vert ^{2} \end{aligned}$$
(4.7)

and

$$\begin{aligned} k_{2} \int _{\Omega } v ( t )\nabla v_{t} ( t )\,dx \leq \frac{C\varepsilon }{2} \bigl\Vert \nabla v ( t ) \bigr\Vert ^{2}+ \frac{k_{2}^{2}}{2\varepsilon } \bigl\Vert v_{t} ( t ) \bigr\Vert ^{2}. \end{aligned}$$
(4.8)

For the source term, we have

$$\begin{aligned} \int _{\Omega }u (t ) f_{1} ( u,v ) \,dx+ \int _{ \Omega }v ( t ) f_{2} ( u,v ) \,dx= (p+1 ) \int _{\Omega }F ( u,v )\,dx. \end{aligned}$$
(4.9)

Combining equations (4.4)–(4.9) and taking \(\varepsilon =\frac{\mu }{2C}\), thus, our proof is completed. □

Lemma 4.3

Let \((u,v,z,y)\) be a solution of problem (2.8). Then the functional

$$\begin{aligned} I ( t ) = \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}}\varrho e^{-\varrho \rho } \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,\rho, \varrho,t ) \bigr\vert ^{2}+ \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,\rho,\varrho,t ) \bigr\vert ^{2} \bigr] \,dx\,d\varrho \,d\rho \end{aligned}$$
(4.10)

satisfies the estimate

$$\begin{aligned} \begin{aligned} I^{\prime } ( t ) \leq {}& -e^{-\tau _{2}} \int _{ \Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,1,\varrho,t ) \bigr\vert ^{2}\,dx\,d\varrho\\ &{} -e^{-\tau _{2}} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,1,\varrho,t ) \bigr\vert ^{2}\,dx\,d\varrho \\ &{}+k_{1} \bigl\Vert u_{t} ( t ) \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{t} ( t ) \bigr\Vert ^{2}-I ( t ). \end{aligned} \end{aligned}$$
(4.11)

Proof

Differentiating (4.10) with respect to t, we get

$$\begin{aligned} \begin{aligned} \frac{d}{dt}I ( t ) ={}&2 \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho e^{-\varrho \rho } \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z ( x,\rho,\varrho,t ) z_{t} ( x,\rho,\varrho,t ) \,dx\,d\varrho \,d\rho \\ &{}+2 \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho e^{- \varrho \rho } \bigl\vert \mu _{2} ( \varrho ) \bigr\vert y ( x,\rho,\varrho,t ) y_{t} ( x,\rho,\varrho,t )\,dx\,d\varrho \,d\rho. \end{aligned} \end{aligned}$$

By using (2.6)–(2.7), we have

$$\begin{aligned} \frac{d}{dt}I ( t ) ={}&{-} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}e^{-\varrho \rho } \biggl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \frac{\partial }{\partial \rho } \bigl\vert z ( x,\rho,\varrho,t ) \bigr\vert ^{2}\\ &{}+ \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \frac{\partial }{\partial \rho } \bigl\vert y ( x,\rho,\varrho,t ) \bigr\vert ^{2} \biggr] \,dx\,d\varrho \,d\rho \\ ={}&{-} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \biggl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \frac{\partial }{\partial \rho } \bigl( e^{-\varrho \rho } \bigl\vert z ( x,\rho,\varrho,t ) \bigr\vert ^{2} \bigr)\\ &{} + \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \frac{\partial }{\partial \rho } \bigl( e^{-\varrho \rho } \bigl\vert y ( x,\rho,\varrho,t ) \bigr\vert ^{2} \bigr) \biggr] \,dx\,d\varrho \,d\rho \\ &{}- \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho e^{- \varrho \rho } \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,\rho, \varrho,t ) \bigr\vert ^{2}+ \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x, \rho,\varrho,t ) \bigr\vert ^{2} \bigr] \,dx\,d\varrho \,d\rho. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{aligned} \frac{d}{dt}I ( t )={}&{-} \int _{\Omega } \int _{ \tau _{1}}^{\tau _{2}}e^{-\varrho } \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,1,\varrho,t ) \bigr\vert ^{2}\,dx\,d\varrho + \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{t} ( x, \varrho,t ) \bigr\Vert ^{2} \\ &{}- \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}}e^{-\varrho } \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,1, \varrho,t ) \bigr\vert ^{2}\,dx\,d\varrho + \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{t} ( x, \varrho,t ) \bigr\Vert ^{2} \\ &{}- \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho e^{- \varrho \rho } \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,\rho, \varrho,t ) \bigr\vert ^{2}+ \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x, \rho,\varrho,t ) \bigr\vert ^{2} \bigr] \,dx\,d\varrho \,d\rho. \end{aligned} \end{aligned}$$

Since \(e^{-\varrho }\) is a deceasing function over \((\tau _{1},\tau _{2} ) \), the desired estimate (4.11) follows immediately from (2.11). □

Proof of Theorem 4.1

Now, we define the following functional:

$$\begin{aligned} \mathcal{F}(t)=NE(t)+\varphi (t)+N_{2}I(t), \end{aligned}$$
(4.12)

where N and \(N_{2} \) are positive constants. It is easy to prove \(\mathcal{F}(t)\) and \(E(t)\) are equivalent, namely there exist two positive constants \(\kappa _{1}\) and \(\kappa _{2}\) such that

$$\begin{aligned} \kappa _{1} E(t)\leq \mathcal{F}(t)\leq \kappa _{2} E(t). \end{aligned}$$
(4.13)

From Lemmas 4.2 and 4.3, we have, for any \(t\geq 0\),

$$\begin{aligned} \mathcal{F}^{\prime } ( t ) \leq{}& -N \biggl[ k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \Vert u_{t} \Vert ^{2}-N \biggl[ k_{2}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \Vert v_{t} \Vert ^{2} \\ &{}+ \biggl(1+\frac{C k_{1}^{2}}{\mu } \biggr) \bigl\Vert u_{t} ( t ) \bigr\Vert ^{2}+ \biggl(1+\frac{Ck_{2}}{\mu } \biggr) \bigl\Vert v_{t} ( t ) \bigr\Vert ^{2}-\frac{\mu }{2} \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}-\frac{\mu }{2} \bigl\Vert \nabla v ( t ) \bigr\Vert ^{2} \\ &{}- ( \lambda +\mu ) \bigl\Vert \operatorname{div} u ( t ) \bigr\Vert ^{2}- ( \lambda +\mu ) \bigl\Vert \operatorname{div} v ( t ) \bigr\Vert ^{2} \\ &{}+\frac{C k_{1}}{\mu } \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert \nabla z ( x,1,\varrho,t ) \bigr\vert ^{2}\,d \varrho\, dx \\ &{}+\frac{Ck_{2}}{\mu } \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert \nabla y ( x,1,\varrho,t ) \bigr\vert ^{2}\,d \varrho\, dx \\ &{}+ (p+1 ) \int _{\Omega }F \bigl( u ( t ),v ( t ) \bigr) \,dx \\ &{}-N_{2}e^{-\tau _{2}} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,1,\varrho,t ) \bigr\vert ^{2}\,dx\,d\varrho \\ &{}-N_{2}e^{- \tau _{2}} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,1, \varrho,t ) \bigr\vert ^{2}\,dx\,d\varrho \\ &{}-N_{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \varrho e^{-\varrho \rho } \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,\rho, \varrho,t ) \bigr\vert ^{2} \\ &{}+ \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,\rho,\varrho,t ) \bigr\vert ^{2} \bigr] \,dx\,d\varrho \,d\rho \\ &{}+N_{2}k_{1} \bigl\Vert u_{t} ( t ) \bigr\Vert ^{2}+N_{2}k_{2} \bigl\Vert v_{t} ( t ) \bigr\Vert ^{2}. \end{aligned}$$
(4.14)

Thus

$$\begin{aligned} \mathcal{F}^{\prime } ( t ) \leq {}&{-} \biggl( N \sigma _{1} - \biggl(1+\frac{C k_{1}^{2}}{\mu } \biggr)-N_{2}k_{1} \biggr) \Vert u_{t} \Vert ^{2}- \biggl( N\sigma _{2} - \biggl(1+\frac{C k_{2}^{2}}{\mu } \biggr)-N_{2}k_{2} \biggr) \Vert v_{t} \Vert ^{2} \\ &{}-\frac{\mu }{2} \bigl\Vert \nabla u ( t ) \bigr\Vert ^{2}- \frac{\mu }{2} \bigl\Vert \nabla v ( t ) \bigr\Vert ^{2}- ( \lambda +\mu ) \bigl\Vert \operatorname{div} u ( t ) \bigr\Vert ^{2}- ( \lambda +\mu ) \bigl\Vert \operatorname{div} v ( t ) \bigr\Vert ^{2} \\ &{}- \biggl( N_{2}e^{-\tau _{2}}-\frac{C k_{1}}{\mu } \biggr) \int _{ \Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho\, dx \\ &{}- \biggl( N_{2}e^{-\tau _{2}}-\frac{Ck_{2}}{\mu } \biggr) \int _{ \Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho\, dx \\ &{}+ (p+1 ) \int _{\Omega }F \bigl( u ( t ),v ( t ) \bigr) \,dx \\ &{}-N_{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \varrho e^{-\varrho \rho } \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z ( x,\rho, \varrho,t ) \bigr\vert ^{2} \\ &{}+ \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y ( x,\rho,\varrho,t ) \bigr\vert ^{2} \bigr] \,dx\,d\varrho \,d\rho, \end{aligned}$$
(4.15)

where

$$\begin{aligned} \sigma _{1}= \biggl[ k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr],\qquad \sigma _{2}= \biggl[ k_{2}- \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d \varrho \biggr) \biggr]. \end{aligned}$$

First, we take \(N_{2}\) large such that

$$\begin{aligned} N_{2}e^{-\tau _{2}}-\frac{C k_{1}}{\mu }>0\quad \text{and}\quad N_{2}e^{-\tau _{2}}- \frac{C k_{2}}{\mu }>0. \end{aligned}$$

We choose \(N > 0 \) large enough so that

$$\begin{aligned} N\sigma _{1} - \biggl(1+\frac{C k_{1}^{2}}{\mu } \biggr)-N_{2}k_{1}>0 \end{aligned}$$

and

$$\begin{aligned} N\sigma _{2} - \biggl(1+\frac{C k_{2}^{2}}{\mu } \biggr)-N_{2}k_{2}>0 \end{aligned}$$

and \(\mathcal{F}\sim E\). Thus we arrive at

$$\begin{aligned} \mathcal{F}^{\prime } (t ) \leq -\alpha \mathcal{F} (t ), \end{aligned}$$

which yields, by integration,

$$\begin{aligned} \mathcal{F} (t )\leq \mathcal{F} (0 )e^{- \alpha t}, \quad\forall t\geq 0. \end{aligned}$$

The use of \(\mathcal{F}\sim E\) gives

$$\begin{aligned} E (t )\leq \beta e^{-\alpha t}, \quad\forall t\geq 0. \end{aligned}$$

 □

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Acknowledgements

The first author would like to thank all the professors of the mathematics department at the University of Annaba in Algeria, especially his Professors/Scientists Pr. Mohamed Haiour, Pr. Ahmed-Salah Chibi, and Pr. Azzedine Benchettah for the important content of masters and PhD courses in pure and applied mathematics which he received during his studies. Moreover, he thanks them for the additional help they provided to him during office hours in their office about the few concepts/difficulties he had encountered, and he appreciates their talent and dedication for their postgraduate students currently and previously. In addition, the idea and research project in this paper was presented by the first author (Pr. Salah Boulaaras) and was carried out by both authors. The authors are grateful to the anonymous referees for the careful reading and their important observations/suggestions for the sake of improving this paper.

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Boulaaras, S., Doudi, N. Global existence and exponential stability of coupled Lamé system with distributed delay and source term without memory term. Bound Value Probl 2020, 173 (2020). https://doi.org/10.1186/s13661-020-01471-9

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MSC

  • 35L90
  • 74G25
  • 35B40
  • 26A51

Keywords

  • Viscoelastic term
  • Delay function
  • Lyapunov functional
  • Distributed delay
  • Exponential decay
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