General enegy decay for a viscoelastic equation of Kirchhoff type with acoustic boundary conditions

TL;DR

This paper investigates the global existence and energy decay of solutions to a viscoelastic Kirchhoff-type equation with acoustic boundary conditions, extending decay results to a broader class of relaxation functions.

Contribution

It establishes global existence and general energy decay for a viscoelastic Kirchhoff equation with acoustic boundary conditions, using a novel approach to relaxation functions.

Findings

Solutions exist globally under certain initial conditions.

Energy decays over time for a wider class of relaxation functions.

The decay rate is characterized using a lemma of P. Martinez.

Abstract

This paper is concerned with a viscoelastic equation of Kirchhoff type with acoustic boundary conditions in a bounded domain of $\mathbb{R}^{n}.$ We show that, under suitable conditions on the initial data, the solution exists globally in time. Then, we prove the general energy decay of global solutions by applying a lemma of P. Martinez, wihch allows us to get our decay result for a class of relaxation functions wider than that usually used.

Equations

$$\left\{\begin{tabular}[]{ll}$u_{tt}-\left(a+b\left\|\nabla u\right\|_{2}^{2\gamma}\right)\Delta u+\int_{0}^{t}g\left(t-s\right)\Delta u\left(s\right)ds=|u|^{k-2}u,$&in $\Omega\times(0,+\infty),$\\ $u=0,$&on $\Gamma_{0}\times(0,+\infty),$\\ $\left(a+b\left\|\nabla u\right\|_{2}^{2\gamma}\right)\frac{\partial u}{\partial\nu}-\int_{0}^{t}g\left(t-s\right)\frac{\partial u}{\partial\nu}\left(s\right)ds=y_{t}$&on $\Gamma_{1}\times(0,+\infty),$\\ $u_{t}+p\left(x\right)y_{t}+q\left(x\right)y=0$&on $\Gamma_{1}\times(0,+\infty),$\\ $u(.,0)=u_{0},u_{t}(.,0)=u_{1},$&in $\Omega$,\\ $y(.,0)=y_{0},$&on $\Gamma_{1}$,\end{tabular}\right.$$

$$\left\{\begin{tabular}[]{ll}$\frac{\partial u}{\partial\nu}=y_{t}$&on $\Gamma\times(0,+\infty),$\\ $\rho u_{t}+m\left(x\right)y_{tt}+p\left(x\right)y_{t}+q\left(x\right)y=0$&on $\Gamma\times(0,+\infty),$\end{tabular}\right.$$

$$\left\{\begin{array}[]{ll}u_{tt}+Lu-\int_{0}^{t}g(t-s)Lu(s)ds=|u|^{k-2}u,&\text{in }\Omega\times(0,\infty),\\ u=0,&\text{on\ }\Gamma_{0}\times\ (0,\infty),\\ \frac{\partial u}{\partial\nu_{L}}-\int_{0}^{t}g(t-s)\frac{\partial u}{\partial\nu_{L}}(s)ds=h(x)y_{t},&\text{on }\Gamma_{1}\times\ (0,\infty),\\ u_{t}+p\left(x\right)y_{t}+q\left(x\right)y=0,&\text{on }\Gamma_{1}\times(0,\infty),\end{array}\right.$$

$$u_{tt}-M\left({\Vert u\Vert }_2^2\right)\Delta u+{\left|u_t\right|}^{\alpha }{u}_t=f(u),$$

$$\left\{\begin{tabular}[]{ll}$u_{tt}-M\left(\left\|\nabla u\right\|_{2}^{2}\right)\Delta u+2\delta u_{t}=0$&in $\Omega\times(0,+\infty),$\\ $u=0,$&on $\Gamma_{0}\times(0,+\infty),$\\ $M\left(\left\|\nabla u\right\|_{2}^{2}\right)\frac{\partial u}{\partial\nu}=y_{t}$&on $\Gamma_{1}\times(0,+\infty),$\\ $u_{t}+p\left(x\right)y_{t}+q\left(x\right)y=0$&on $\Gamma_{1}\times(0,+\infty),$\end{tabular}\right.$$

$$\left\{\begin{tabular}[]{ll}$\left|u_{t}\right|^{\rho}u_{tt}-\left(M\left(u\right)\left(t\right)\right)\Delta u-\Delta u_{tt}$&\\ \ \ \ \ \ $+\int_{0}^{t}g\left(t-s\right)\Delta u\left(s\right)ds+|u_{t}|^{q}u_{t}=|u|^{p}u$&in $\Omega\times(0,+\infty),$\\ $u=0,$&on $\Gamma_{0}\times(0,+\infty),$\\ $\left(M\left(u\right)\left(t\right)\right)\frac{\partial u}{\partial\nu}+\frac{\partial u_{tt}}{\partial\nu}-\int_{0}^{t}g\left(t-s\right)\frac{\partial u}{\partial\nu}\left(s\right)ds$&\\ \ \ \ \ \ \ \ \ $+\mu_{0}u_{t}\left(x,t\right)+\mu_{1}u_{t}\left(x,t-\tau\left(t\right)\right)=h\left(x\right)y_{t}$&on $\Gamma_{1}\times(0,+\infty),$\\ $u_{t}+p\left(x\right)y_{t}+q\left(x\right)y=0$&on $\Gamma_{1}\times(0,+\infty),$\\ $u_{t}\left(x,t-\tau\left(t\right)\right)=f_{0}(x,t)$&on $\Gamma_{1},-\tau\left(0\right)\leq t\leq 0$,\end{tabular}\right.$$

$${\Vert u\Vert }_k\le C_{\ast }{\Vert \nabla u\Vert }_2,\text{ for every }u\in V,\text{and }1\le k\le \overline{k}$$

$${\Vert u\Vert }_{2,{\Gamma }_1}\le {\overline{C}}_{\ast }{\Vert \nabla u\Vert }_2,\text{ for every }u\in V,$$

$$0<p_0\le p(x)\le p_1;0<q_0\le q(x)\le q_1,\text{for all }x\in {\Gamma }_1.$$

$$g\left(0\right)>0\text{ and }a-{\int }_0^{\infty }g\left(s\right)ds=l>0.$$

$${\int }_0^{\infty }\xi (s)ds=+\infty ,g^{\prime }(t)\le -\xi (t)g(t),\text{ for all }t\ge 0,.$$

$$\frac{{\xi }^{\prime }}{{\xi }^{\theta }}\in L^1\left(0,+\infty \right),{\int }_t^{t+s}\frac{{\xi }^{\prime }(\tau )}{\xi (\tau )}d\tau \le r,\text{ for all }t,s\ge 0,.$$

$$u\in C(0,T;V),$$

$$E\left(t\right)=\frac{1}{2}{\Vert u_t\Vert }_2^2+J\left(u,y\right)\left(t\right),$$

$$\begin{gathered} J\left(u,y\right)\left(t\right)=\frac{1}{2}\left(a-\left({\int }_0^{t}g\left(s\right)ds\right)\right){\Vert \nabla u\Vert }_2^2+\frac{b}{\gamma +1}{\Vert \nabla u\Vert }_2^{2\left(\gamma +1\right)} \\ +{\int }_{{\Gamma }_1}\frac{1}{2}\left(q\left(x\right)y^2\right)dx+\frac{1}{2}\left(g\diamond \nabla u\right)\left(t\right)-\frac{1}{k}{\Vert u\Vert }_k^k, \end{gathered}$$

$$\left(g\diamond \nabla u\right)\left(t\right)={\int }_0^{t}g\left(t-s\right){\Vert \nabla u\left(t\right)-\nabla u\left(s\right)\Vert }_2^{2}ds.$$

$$E^{\prime }\left(t\right)=-\frac{1}{2}g\left(t\right){\Vert \nabla u\Vert }_2^2+\frac{1}{2}\left(g^{\prime }\diamond \nabla u\right)\left(t\right)-{\int }_{{\Gamma }_1}p\left(x\right)y_t^{2}dx\text{, for all }t\ge 0.$$

$$\begin{gathered} \frac{d}{dt}(\frac{1}{2}{\Vert u_t\Vert }_2^2+\frac{1}{2}a{\Vert \nabla u\Vert }_2^2+\frac{b}{2\left(\gamma +1\right)}{\Vert \nabla u\Vert }_2^{2\left(\gamma +1\right)}-\frac{1}{2}\left({\int }_0^{t}g\left(s\right)ds\right){\Vert \nabla u\Vert }_2^2 \\ +\frac{1}{2}\left(g\diamond \nabla u\right)\left(t\right)-\frac{1}{k}{\Vert u\Vert }_k^k)={\int }_{{\Gamma }_1}{u}_t{y}_{t}dx-\frac{1}{2}g\left(t\right){\Vert \nabla u\Vert }_2^{2}dx+\frac{1}{2}\left(g^{\prime }\diamond \nabla u\right)\left(t\right) \end{gathered}$$

$${\int }_{{\Gamma }_1}{u}_t{y}_{t}dx=-{\int }_{{\Gamma }_1}p\left(x\right)y_t^{2}dx-\frac{d}{dt}\left({\int }_{{\Gamma }_1}\frac{1}{2}\left(q\left(x\right)y^2\right)dx\right).$$

$$F(x)=\frac{1}{2}{x}^2-\frac{B_{\Omega }^k}{k}{x}^k$$

$$B_{\Omega }=\underset{u\in V,u\ne 0}{\sup }\frac{{\Vert u\Vert }_k}{\sqrt{l{\Vert \nabla u\Vert }_2^2+\frac{b}{\gamma +1}{\Vert \nabla u\Vert }_2^{2\left(\gamma +1\right)}}}$$

$${\lambda }_1=B_{\Omega }^{\frac{-k}{k-2}}.$$

$$d_1=F({\lambda }_1)=\left(\frac{k-2}{2k}\right)B_{\Omega }^{\frac{-2k}{k-2}}$$

$$\begin{gathered} E\left(t\right)\ge J\left(u,y\right)\left(t\right)=\frac{1}{2}\left(a-\left({\int }_0^{t}g\left(s\right)ds\right)\right){\Vert \nabla u\Vert }_2^2+\frac{b}{2\left(\gamma +1\right)}{\Vert \nabla u\Vert }_2^{2\left(\gamma +1\right)} \\ +{\int }_{{\Gamma }_1}\frac{1}{2}\left(q\left(x\right)y^2\right)dx+\frac{1}{2}\left(g\diamond \nabla u\right)\left(t\right)-\frac{1}{k}{\Vert u\Vert }_k^k \\ \ge \frac{1}{2}{\gamma }^2\left(t\right)-\frac{1}{k}{B}_{\Omega }^k{\gamma }^k\left(t\right)=F\left(\gamma \left(t\right)\right) \end{gathered}$$

$$\gamma \left(t\right)=\sqrt{l{\Vert \nabla u\Vert }_2^2+\frac{b}{\left(\gamma +1\right)}{\Vert \nabla u\Vert }_2^{2\left(\gamma +1\right)}+{\int }_{{\Gamma }_1}q\left(x\right)y^{2}dx+\left(g\diamond \nabla u\right)\left(t\right)}.$$

$$\begin{gathered} E\left(t\right)\ge F\left(\gamma \left(t\right)\right)={\gamma }^2\left(t\right)\left(\frac{1}{2}-\frac{1}{k}{B}_{\Omega }^k{\gamma }^{k-2}\left(t\right)\right) \\ >{\gamma }^2\left(t\right)\left(\frac{1}{2}-\frac{1}{k}{B}_{\Omega }^k{\lambda }_1^{k-2}\right) \\ >\frac{k-2}{2k}{\gamma }^2\left(t\right). \end{gathered}$$

$$E(t)\le E(0)<d_1\text{and }F\left(\gamma \left(t\right)\right)<d_1\text{for all }t\in [0,T).$$

$$\begin{gathered} \frac{1}{2}{\Vert u_t\Vert }_2^2+\frac{k-2}{2k}{\gamma }^2\left(t\right)\le \frac{1}{2}{\Vert u_t\Vert }_2^2+F\left(\gamma \left(t\right)\right) \\ \le \frac{1}{2}{\Vert u_t\Vert }_2^2+J\left(u,y\right)\left(t\right)=E(t)\le E(0)<d_1, \end{gathered}$$

$$\varphi (0)=0\text{and}\varphi (t)\to +\infty \text{as}t\to +\infty .$$

$${\int }_S^{+\infty }{E}^{1+\sigma }(t){\varphi }^{\prime }(t)dt\le \frac{1}{\omega }{E}^{\sigma }(0)E(S).\text{for all }0\le S<+\infty ,$$

Full text

Keywords and phrases : Kirchhoff type equation, Energy decay, Acoustic boundary conditions, Memory term, Multiplier method. 

**AMS Classification : ** 35L72, 35A01, 35B40.

1 Introduction

In this paper we are concerned with the decay rates of solutions for the following nonlinear wave equation of Kirchhoff type, with acoustic boundary conditions

[TABLE]

where $\Omega$ is a bounded domain of $\mathbb{R}^{n},$ $n\geq 1$ having a smooth boundary $\Gamma=\partial\Omega$, consisting of two closed and disjoint parts $\Gamma_{0}$ and $\Gamma_{1}$. Here $\nu$ denotes the unit outward normal to $\Gamma$. The parameters $a>0,\ b>0,$ $\gamma\geq 0$ and $k>2$ are constant real numbers, p\and $q$ are given functions satisfying some conditions to be specified later; $u_{0},u_{1}:\Omega\rightarrow\mathbb{R}$ and $\ y_{0}:\Gamma_{1}\rightarrow\mathbb{R}$ are given functions.

The acoustic boundary conditions were introduced by Beale and Rosencrans in [6], [7]; where the authors proved the global existence and regularity of solutions of the wave equation subject to boundary conditions of the form

[TABLE]

where $\rho>0$, $p$ is a nonnegative function and $m$, $q$ are strictly positive functions on the boundary.

Wave equations with acoustic boundary conditions have been treated by many authors [4], [5], [2], [8], [19], [9] . In [4], the authors studied a linear wave equation subject to the boundary conditions (1.2), with $m\equiv 0$, on the portion $\Gamma_{1}$ of the boundary and Dirichlet boundary conditions on the portion $\Gamma_{0}.$ They proved global solvability and uniform energy decay.

Boukhatem and Benabderrahmane [19], studied the variable-coefficient viscoelastic wave equation

[TABLE]

where $Lu=-div(A\nabla u)$, $\frac{\partial u}{\partial\nu_{L}}=\left(A\nabla u\right).\nu$ and $A=\left(a_{ij}(x)\right)$ a matrix with $a_{ij}\in C^{1}(\overline{\Omega})$. Combining the techniques used by Georgiev and Todorova in [17], those used by Frota and Larkin in [4], and Faedo–Galerkin’s approximations, the authors proved global solvability for suitable initial data, and general energy decay for some relaxation functions $g$ satifaying some known condtions, introduced firstly by Messaoudi in [14].

For quasilinear equations, the authors of [3] studied the Carrier equation

[TABLE]

subject to the boundary conditions (1.2) on the portion $\Gamma_{1}$ and Dirichlet boundary conditions on the portion $\Gamma_{0},$ where $M:\mathbb{R}_{+}\longrightarrow\mathbb{R}$ is a $C^{1}$ function satisfying some conditions. They proved the existence and uniqueness of global solutions. In [20], the author studied the following problem

[TABLE]

where $M\left(\left\|\nabla u\right\|_{2}^{2}\right)=a+b\left\|\nabla u\right\|_{2}^{2}$, with $a,b>0.$ He proved the uniform stability of solutions for a sufficiently small positif passive viscous damping coeficient $\delta$, using multiplier technique.

Recently, Lee et all in [9] were concerned with the following Kirchhoff type equation with Balakrishnan-Taylor damping, time-varying delay and boundary conditions:

[TABLE]

where $M\left(u\right)\left(t\right)=a+b\left\|\nabla u\right\|_{2}^{2}+\sigma\left(\nabla u,\nabla u_{t}\right)$, with $a,b,\sigma>0.$ They showed the global existence of solutions and established a general energy decay

Motivated by the previous works, we consider a non degenerate Kirchhoff type wave equation with memory term, acoustic boundary conditions and source term. This model is new because has not been considered by predecessors. Further, we do not use differetial inequalities to prove our decay result, but we use a method based on integral inequalities, introduced by P. Martinez [13] (Lemma 4.1 below) and that generalize those introduced by V. Komornik [18] and A. Haraux [1]. This method allows us to consider a class of relaxation functions larger than the one usually considered.

The paper is organized as follows. In Section 2, we present some notations and material needed for our work and state an existence result, which can be proved by combining the techniques used in [10] and [19]. In section 3, we prove that, for suitable initial data, the solution exists globally in time. In this section, our proof technique follows the arguments of [16], with some modifications being needed for our problem. Section 4 contains the statement and the proof of our main result.

2 Preliminaries and some notations

In this section, we present some notations and some material needed in the proof of our result.

Let $V=\left\{u\in H^{1}\left(\Omega\right):u=0\text{ on }\Gamma_{0}\right\}$ be the subspace of the classical Sobolev space $H^{1}\left(\Omega\right)$ of real valued functions of order one. The Poincare’s inequality holds on $V$, i.e. there exists a positive constant $C_{\ast}$ (depends only on $\Omega$) such that:

[TABLE]

where $\left\|u\right\|_{k}^{k}=\int_{\Omega}\left|u\right|^{k}dx$, $\left\|\nabla u\right\|_{2}^{2}=\int_{\Omega}\left|\nabla u\right|^{2}dx$ and \overline{k}=\left\{\begin{array}[]{l}\frac{2n}{n-2}~{}\text{, if }n\geq 3\\ +\infty~{}\text{, if }n=1\text{ or }2\end{array}\right..

According to the trace theory, there exists a positive constant $\overline{C}_{\ast}$ (depends only on $\Gamma_{1}$) such that:

[TABLE]

where $\left\|u\right\|_{2,\Gamma_{1}}^{2}=\int_{\Gamma_{1}}\left|u\right|^{2}dx$.

To prove our result, we need the following assumptions.

(H1) For the functions $p$ and $q$, we assume that $p,q\in C(\Gamma_{1})$ and $p(x)>0$, $q(x)>0$, for all $x\in\Gamma_{1}$. This assumption assures us that there exist positive constants $p_{i},q_{i}\ (i\in\left\{0,1\right\})$ such that:

[TABLE]

(H2) For the relaxation function $g:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, we assume that :

[TABLE]

and there exists a locally absolutely continuous function  $\xi:\left[0,+\infty\right)\rightarrow\left(0,+\infty\right)$, and constants $\theta\geq 0,r$ such that

[TABLE]

[TABLE]

The last assumption assures us that $\xi\left(t+s\right)\leq e^{r}\xi\left(t\right),$ for all $t,s\geq 0,$ and so $\xi\in L^{\infty}\left(0,+\infty\right).$

Remark 2.1

(i) Notice that, if $\xi$ is a nonincreasing function, then hypothesis (2.6) is trivially valid with $\theta=$ $r=0$.

(ii) Notice also that, proofs in previous papers depend strongly on the nonincreasence of $\xi.$ We prove our result without this restriction.

We now state a local existence theorem for problem (1.1), whose proof follows the arguments in [19].

Theorem 2.2** (Local existence)**

Let $2<k\leq\frac{2n-2}{n-2}$ , $(u_{0},u_{1})\in$ $V\times L^{2}(\Omega)$   and $y_{0}\in L^{2}(\Gamma_{1})$. Suppose that hypotheses (H1)–(H2) hold, then there exists a unique pair of functions $(u,y)$, which is a solution of the problem (1.1) such that

[TABLE]

for some $T>0$.

3 Global existence

In this section, we shall state and prove the global existence. For this purpose, we define the energy of problem (1.1) by

[TABLE]

where

[TABLE]

and

[TABLE]

Lemma 3.1

Let $(u,y)$ be the solution of (1.1). Then, the energy functional defined by (3.1) is a nonincreasing function and

[TABLE]

Proof. Multiplying the first equation in (1.1) by $u_{t}$, integrating over $\Omega$, using integration by parts and exploiting the third equation in system (1.1), we get

[TABLE]

From the fourth equation in (1.1), we deduice that

[TABLE]

Plugging (3.6) into (3.5) and making use of (3.1), then we obtain (3.4), and hence E^{\prime}\left(t\right)\leq 0,\ \for all $t\geq 0$.

As in [11] and [16], we define a functional $F$, which helps in establishing the global existence of solution.

Setting

[TABLE]

where

[TABLE]

We can verify, as in [11], that the function $F$ is increasing in $\left(0,\lambda_{1}\right)$ and decreasing in $\left(\lambda_{1},+\infty\right),$ where $\lambda_{1}$ is the strictly positive zero of the derivative function $F^{\prime},$ that is

[TABLE]

$F$ has a maximum at $\lambda_{1}$ with the maximum value

[TABLE]

From (3.1), (3.2), (3.8), (2.4) and (3.7), we have

[TABLE]

where

[TABLE]

If $\gamma(t)<\lambda_{1}$, then we get

[TABLE]

Lemma 3.2

*Let $2<k\leq\frac{2n-2}{n-2}$, $(u_{0},u_{1})\in$ $V\times L^{2}(\Omega)$, $y_{0}\in L^{2}(\Gamma_{1})$ and hypotheses (H1)–(H2) hold. Assume further that $E\left(0\right)<d_{1}$ and

$\gamma\left(0\right)=\sqrt{l\left\|\nabla u_{0}\right\|_{2}^{2}+\frac{b}{\left(\gamma+1\right)}\left\|\nabla u_{0}\right\|_{2}^{2\left(\gamma+1\right)}+\int_{\Gamma_{1}}q\left(x\right)y_{0}^{2}dx}<\lambda_{1}.$ Then

$E\left(t\right)<d_{1}$

 

and $\gamma\left(t\right)=\sqrt{l\left\|\nabla u\right\|_{2}^{2}+\frac{b}{\left(\gamma+1\right)}\left\|\nabla u\right\|_{2}^{2\left(\gamma+1\right)}+\int_{\Gamma_{1}}q\left(x\right)y^{2}dx+\left(g\diamond\nabla u\right)\left(t\right)}<\lambda_{1},$ for all $t\in[0,T)$.*

Proof. Using the fact that $E(t)$ is a non-increasing function and (3.11) , we get

[TABLE]

From (3.12) and the fact that $E(0)<d_{1}$, it follows that there exist $\lambda_{0},\lambda_{2}$ such that $0<\lambda_{0}<\lambda_{1}<\lambda_{2}$ and $F(\lambda_{0})=E(0)=F(\lambda_{2}).$ As $F\left(\gamma\left(0\right)\right)\leq E(0)=F(\lambda_{0})$ and $\lambda_{0},\gamma\left(0\right)\in$ $\left(0,\lambda_{1}\right)$ where the function $F$ is increasing, then $\gamma\left(0\right)\leq\lambda_{0}.$

We argue by contradiction to prove that $\gamma\left(t\right)\leq\lambda_{0}$,  for all $t\in[0,T).$

Suppose that there exists $t^{\ast}\in(0,T)$ such that $\gamma(t^{\ast})>\lambda_{0}.$ We have two cases.

Case 1: $\lambda_{0}<\gamma(t^{\ast})<\lambda_{1},$ then, by virtue of the increasince of $F$ and the non-increasince of $E$, we get

$F\left(\gamma(t^{\ast})\right)>F\left(\lambda_{0}\right)=E(0)\geq E(t^{\ast}),$ which contradicts (3.11).

Case 2: $\lambda_{1}\leq\gamma(t^{\ast}),$ then, by virtue of the continuity of $\gamma$ on $(0,t^{\ast}),$ there exists $t_{0}\in\left(0,t^{\ast}\right)$ such that $\lambda_{0}<\gamma(t_{0})<\lambda_{1},$ which yields a contradiction as in case 1.

Thus, $\gamma\left(t\right)\leq\lambda_{0}$, and so \gamma\left(t\right)\leq\lambda_{1}\for all $t\in[0,T).$

Theorem 3.1

Let $\left(u\left(t\right),y\left(t\right)\right)$ be the solution of (1.1). If $(u_{0},u_{1})\in$ $V\times L^{2}(\Omega),\ y_{0}\in L^{2}(\Gamma_{1})$ such that, $\gamma\left(0\right)=\sqrt{l\left\|\nabla u_{0}\right\|_{2}^{2}+\frac{b}{\left(\gamma+1\right)}\left\|\nabla u_{0}\right\|_{2}^{2\left(\gamma+1\right)}+\int_{\Gamma_{1}}q\left(x\right)y_{0}^{2}dx}<\lambda_{1}$ and $E\left(0\right)<d_{1}$, then the solution $\left(u\left(t\right),y\left(t\right)\right)$ is global in time.

Proof. Using Lemma 3.2, (3.12), (3.11) and (3.1), we get

[TABLE]

which ensures the boundedness of $u$ in $V,$ $u_{t}$ in $L^{2}(\Omega)$ and $y$ in $L^{2}(\Gamma_{1}),$ and hence the solution $\left(u\left(t\right),y\left(t\right)\right)$ is bounded and global in time.

4 Decay result

In order to study the decay estimate of global solution for the problem (1.1), we need the following lemma.

Lemma 4.1

*([13]) Let $E:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be a non-increasing function and

$\phi:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ a strictly increasing $C^{1}$ function such that*

[TABLE]

Assume that there exist $\sigma\geq 0$ and $\omega>0$ such that

[TABLE]

then

[TABLE]

[TABLE]

Our main result is the following

Theorem 4.1

Under assumptions of Theorem 2.2 and the assumption that $E\left(0\right)<d_{1}$ and $\gamma\left(0\right)<\lambda_{1}$, there exists a positive constant $\omega$ depending on initial energy $E(0)$, such that the solution energy of (1.1) satisfies,

[TABLE]

Remark 4.2

If $E(t_{0})=0$, for some $t_{0}\geq 0$; then from Lemma 3.1, we have $E(t)=0,$ for all $t\geq t_{0}$, and then we have nothing to prove in this case. So we assume that $E(t)>0,$ for all $t\geq 0$ without loss of generality.

Proof. Since $g$ is positive and $g(0)>0$ then for any $t_{0}>0$ we have

[TABLE]

Multiplying the first equation in (1.1) by $\xi(t)u(t)$ and integrating by parts over $\Omega\times(S,T),$ with $S\geq t_{0}$, we get

[TABLE]

since $\gamma\geq 0,$ we deduce that

[TABLE]

Recalling the definition of $B_{\Omega}$ above and using (3.12), we can estimate

[TABLE]

In view of this inequality there exists a constant $c>0$ independent of $t$, such that

[TABLE]

Adding some terms to both sides of the obove inequality and using (4.3), we deduce

[TABLE]

Using the Cauchy-Schwarz’s inequality, the Poincaré’s inequalities (2.1), (2.4) and the definition of energy (3.1), we obtain estimates as follows

[TABLE]

Now, using Cauchy’s inequality, (3.1), (2.4) and a technique used in [14], we get

[TABLE]

for any $\epsilon>0.$

The following estimate is obtained using the trace theory, hypotesis (2.3), Poincaré’s inequalitie (2.2), Hölder’s and Cauchy’s inequalities

[TABLE]

for any $\epsilon>0,$

To estimate the sixth term of (4.6), we multiply the fourth equation in (1.1) by $\xi(t)y$, integrate by parts over $\left(S,T\right)\times\Gamma_{1}$ and use Cauchy’s inequalitie:

[TABLE]

Using hypotesis (2.3), and (3.4), we get

[TABLE]

Using the trace theory, (2.2), (3.1), Hölder’s and Cauchy’s inequalities, we get

[TABLE]

Combining now (4.9), (4.11), (4.12) and the fact that $E$ is decreasing and $\xi$ is bounded, we get

[TABLE]

where c_{0}\left(\epsilon\right)=4\left(\left\|\xi\right\|_{\infty}\frac{k}{\left(k-2\right)}\left(\frac{\overline{C}_{\ast}^{2}}{q_{0}l}+1\right)+\frac{k}{\left(k-2\right)}\left(\frac{\overline{C}_{\ast}^{2}}{q_{0}l}+1\right)\left\|\xi\right\|_{\infty}^{\theta}\left\|\frac{\xi^{\prime}}{\xi^{\theta}}\right\|_{1}+\left\|\xi\right\|_{\infty}\frac{p_{1}}{q_{0}}+\left\|\xi\right\|_{\infty}\frac{1}{\epsilon}\frac{1}{p_{0}}\right)\

and $c_{1}=\overline{C}_{\ast}^{2}\frac{4k}{\left(k-2\right)l}.$

To estimate the third term of (4.6), we multiply the first equation in (1.1) by $\xi(t)\int_{0}^{t}g(t-s)(u(t)-u(s))ds$, integrate by parts over $\Omega\times(S,T)$ and utilize (4.1):

[TABLE]

As above, we can obtain the following estimates

[TABLE]

[TABLE]

and

[TABLE]

Combining the estimate (4.17), the fact that $\left\|\nabla u\right\|_{2}^{2}\leq\frac{2k}{\left(k-2\right)l}E\left(t\right)$ and again $\int_{0}^{t}g(s)ds\leq a-l,$ we get

[TABLE]

The trace theory, hypotesis (2.3), Poincaré’s inequalitie (2.2), Hölder’s and Cauchy’s inequalities, permit us to get

[TABLE]

Using the fact that $2<k\leq\frac{2n-2}{n-2}$ and Poincaré’s inequalitie (2.1), the last term on the right hand side of (4.14), can be estimated as follows

[TABLE]

Combining estimates (4.14)-(4.20), we get

[TABLE]

where $c_{2}\left(\epsilon\right)=g_{0}^{-1}\left(\left(2\xi_{\infty}+\left\|\xi\right\|_{\infty}^{\theta}\left\|\frac{\xi^{\prime}}{\xi^{\theta}}\right\|_{1}\right)\left(1+\left(a-l\right)C_{\ast}^{2}\frac{k}{k-2}\right)+\frac{\xi_{\infty}}{2}\left(\frac{C_{\ast}^{2}}{\epsilon}g(0)+\frac{1}{p_{0}}\right)\right),$

$c_{3}=g_{0}^{-1}\left(2+\frac{2k\left(2a-l+b\left(\frac{2k}{\left(k-2\right)l}E\left(0\right)\right)^{\gamma}\right)}{\left(k-2\right)l}+\left(\frac{2kC_{\ast}^{2}}{\left(k-2\right)l}\right)^{k-1}\left(E\left(0\right)\right)^{k-2}\right)$ and

$c_{4}\left(\epsilon\right)=g_{0}^{-1}\left(\frac{\left(2a-l+b\left(\frac{2k}{\left(k-2\right)l}E\left(0\right)\right)^{\gamma}\right)\left(a-l\right)}{4\epsilon}+\left(a-l\right)\left(1+\frac{1}{2}\overline{C}_{\ast}^{2}+\frac{1}{4\epsilon}C_{\ast}^{2}\right)\right).$

From hypotesis (2.5) and (2.6), it follows that

[TABLE]

Consequently, using (4.6), (4.7), (4.8), (4.21), (4.9), (4.13), (4.22), (2.6) and the fact that $E$ is decreasing , we get

[TABLE]

where $c_{5}\left(\epsilon\right)=\sqrt{\frac{k-2}{lk}}\left(2\left\|\xi\right\|_{\infty}C_{\ast}+C_{\ast}\left\|\xi\right\|_{\infty}^{\theta}\left\|\frac{\xi^{\prime}}{\xi^{\theta}}\right\|_{1}\right)+c_{2}\left(\epsilon\right)\left(1+\frac{c}{2}\right)+\left(\frac{1}{2\epsilon}\left(a-l\right)+2c_{4}\left(\epsilon\right)\left(1+c\right)\right)e^{r}$

$\ \ \ \ \ \ \ \ \ \ \ +\frac{1}{\epsilon}\frac{1}{p_{0}}\left\|\xi\right\|_{\infty}+\frac{c}{2}c_{0}\left(\epsilon\right)$ and $c_{6}=\frac{2k}{\left(k-2\right)l}\left(1+\overline{C}_{\ast}^{2}\right)+c_{3}+\left(c_{1}+c_{3}\right)\frac{c}{2}$

and so, for $\epsilon$ small enough, we get

[TABLE]

and if $0\leq S<t_{0},$ it suffices to observe that

[TABLE]

therefore \int_{S}^{T}\xi(t)E\left(t\right)dt\leq CE\left(S\right),\for all S\geq 0\and some constant $C\geq 0$ independent of $S$ and $T$

Let $T\rightarrow+\infty,$ applying Lemma 4.1 (with $\sigma=0$ and $\phi\left(t\right)=\int_{0}^{t}\xi(\tau)d\tau$), we conclude that $E(t)\leq E(0)e^{1-\omega\int_{0}^{t}\xi(\tau)d\tau},$ for all $t\geq 0,$ for some $\omega=\omega\left(E\left(0\right),\xi;t_{0}\right)$