\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 160, pp. 1--19.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/160\hfil Stabilization of wave equations] {Stabilization of wave equations with variable coefficients and internal memory} \author[Z.-H. Ning, F. Yang \hfil EJDE-2018/160\hfilneg] {Zhen-Hu Ning, Fengyan Yang} \address{Zhen-Hu Ning \newline Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China} \email{ningzhenhu@bjut.edu.cn} \address{Fengyan Yang \newline Key Laboratory of Systems and Control, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China} \email{yangfengyan12@mails.ucas.ac.cn} \dedicatory{Communicated by Goong Chen} \thanks{Submitted July 21, 2018. Published September 5, 2018.} \subjclass[2010]{93C20, 93D20} \keywords{Stabilization; wave equation with variable coefficients; memory term; \hfill\break\indent time-varying delay; geometric conditions} \begin{abstract} In this article, we consider the stabilization of a wave equation with variable coefficients and internal memory in an open bounded domain, by the Riemannian geometry approach. For the wave equation with a locally distributed memory with a kernel, we obtain exponential decay of the energy under some geometric conditions. In addition, for the wave equation with nonlinear internal time-varying delay without upper bound, we obtain uniform decay of the energy. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction and statement of main results} Let $\Omega$ be an open bounded domain in $ \mathbb{R}^n$ with smooth boundary $\Gamma$. Define \begin{equation} \mathscr{A} u=-\operatorname{div} A(x)\nabla u\quad\text{for } u\in H^1(\Omega)\label{mu6}, \end{equation} where $A(x)=(a_{ij}(x))_{n\times n}$ is a symmetric, positively definite matric for each $x\in\mathbb{R}^n$ and $a_{ij}(x)$ is a smooth function on $\mathbb{R}^n$ for each $1\leq i,j \leq n$. We consider the stabilization of the problem \begin{equation}\label{e1.1} \begin{gathered} u_{tt}+\mathscr{A} u+a(x)[\mu_1u_t(x,t) +\mu_2 \int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho]=0\\ (x,t)\in \Omega\times (0,+\infty),\\ u(x,t)\big|_{\Gamma}=0\quad t\in(0,+\infty),\\ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x)\quad x\in \Omega,\\ u_t(x,t)=f_0(x,t)\quad (x,t)\in \Omega\times (-\infty,0), \end{gathered} \end{equation} where $a(x)\in C^1(\overline{\Omega})$ is a nonnegative function and the kernel $k(\cdot)$ satisfies \begin{equation} \label{e2.210}\int_0^{\infty}|k(\rho)|d\rho=1. \end{equation} Moreover, $\mu_1,\mu_2>0$, and the initial data $(u_0,u_1, f_0)$ belongs to a suitable space. Stability results for system \eqref{e1.1} in the case of $\mu_2 = 0$; that is, without memory, were obtained by some authors. See \cite{ZY3,ZY21,ZY23}. Time delays often arise in many physical, chemical, biological and economical phenomena. In recent years, different equations with time delay effects have become an active area of research. In particular, as is well-known that an arbitrarily small delay may be the source of instability and some dissipative mechanism need to be introduced to against the instabilities, the control and stabilization of the wave equations with time delay have been extensively studied by several authors (see for example, \cite{ZY11,ZY10,ZY17,ZY31,NP,ZY5,ZY8,ZY9,ZY13,ZY16,ZY1,ZY12} and many others.) To be specific, with a internal or boundary constant delay term, the stability and instability results of the constant coefficient wave equation are given by \cite{ZY11,NP,ZY12}. The results in \cite{NP} have been extended to the variable coefficient wave equation in \cite{ZY13,ZY1}. Besides, with a time-varying delay term in the boundary or interior, the uniform decay results of the energy of the constant coefficient wave equation are obtained by \cite{ZY10,ZY17,ZY31,ZY5,ZY8,ZY9,ZY16}. The following system was studied in \cite{ZY14}. \begin{equation}\label{exx.0} \begin{gathered} u_{tt}+\mathscr{A} u=0\quad (x,t)\in \Omega\times (0,+\infty),\\ u(x,t) =0\quad (x,t)\in \Gamma_2 \times (0,+\infty),\\ \frac{\partial u}{\partial \nu_{\mathscr{A}}} +bu_t(x,t)+ \int_0^{t}k(t-\rho) u_t(x,\rho)d\rho=0\quad (x,t)\in \Gamma_1 \times (0,+\infty),\\ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x)\quad x\in \Omega,\\ u_t(x,t)=f_0(x,t)\quad (x,t)\in \Gamma_1\times (-\infty,0) \end{gathered} \end{equation} where $\Gamma=\Gamma_1\cup\Gamma_2$, $\overline\Gamma_1\cap\overline\Gamma_2=\emptyset$ and $\Gamma_2\not=\emptyset$. $b$ is a positive constant and $\frac{\partial u}{\partial \nu_{\mathscr{A}}}=\langle A(x)u,\nu\rangle $ is the co-normal derivative, $\langle \cdot,\cdot\rangle $ denotes the standard metric of the Euclidean space $\mathbb{R}^n$ and $\nu(x)$ is the outside unit normal vector for each $x\in\Gamma$. The exponential decay of the energy of system \eqref{exx.0} is obtained under the following assumptions: the kernel $k(\cdot)$ satisfies \begin{equation} k(t)\geq 0, \quad k'(t)\leq -\gamma_0 k(t),\quad k''(t)\geq- \gamma_1 k'(t), \end{equation} where $\gamma_0, \gamma_1$ are positive constants, and there exists a vector field $H$ on $\overline{\Omega}$ and a constant $\rho_0>0$ such that \begin{gather} D_gH(X,X)\geq \rho_0|X|_g^2\quad\text{for}\quad X\in\mathbb{R}^n_x\quad x\in\overline{\Omega}, \\ \sup_{x\in \overline{\Omega}}\operatorname{div} H < \inf_{x\in \overline{\Omega}} \operatorname{div} H +2\rho_0, \\ H\cdot \nu \leq0 \quad x\in \Gamma_2 \quad \text{and} \quad H\cdot \nu \geq\delta\quad x\in \Gamma_1, \end{gather} where $\delta$ is a positive constant. Note that the initial memory of $u_t$ is zero and $\int_0^{t}k(t-\rho) u_t(x,\rho)d\rho=\int_0^{-t}k(\rho)$ $u_t(x,t+\rho)d\rho$ in \eqref{exx.0}. Our objective in this paper is to study the exponential decay of the energy of system \eqref{e1.1} with a nonzero initial memory of $u_t$, a more general kernel $k(\cdot)$ and vector field $H$ than \eqref{exx.0}. To obtain our stabilization result, we assume that \begin{equation} \mu_2<\mu_1.\label{mu1} \end{equation} Let \begin{equation}\label{e2.211} G(h)=\int_{h}^{+\infty}|k(\rho)|d\rho\ \ for \ \ h\geq0. \end{equation} Define the energy of system \eqref{e1.1} by \begin{equation} \label{e2.1} \begin{aligned} E(t)&=\frac12\int_{\Omega}\Big(u_t^2+\sum_{i,j=1}^na_{ij}u_{x_i}u_{x_j}\Big)dx\\ &\quad +\xi\int_{0}^{\infty} \int_{\Omega}a(x)G(\rho)u^2_t(x,t-\rho)\,dx\,d\rho, \end{aligned} \end{equation} where $\xi$ is a positive constant satisfying \begin{equation} \mu_2< 2\xi<2\mu_1-\mu_2.\label{mu2} \end{equation} As in \cite{Y, ZY22}, we define \begin{equation}\label{g} g=A^{-1}(x)\quad\text{for } x\in\mathbb{R}^n \end{equation} as a Riemannian metric on $\mathbb{R}^n$ and consider the couple $(\mathbb{R}^n,g)$ as a Riemannian manifold. For each $x\in\mathbb{R}^n$, the metric $g$ introduces an inner product and the norm on the tangent space on $\mathbb{R}^n_x=\mathbb{R}^n$ by $$ \langle X,Y\rangle _g=\langle A^{-1}(x)X,Y\rangle ,\quad |X|_g^2=\langle X,X\rangle _g\quad X,\,Y\in\mathbb{R}^n_x, $$ where $\langle \cdot,\cdot\rangle $ denotes the standard dot metric. Let $f\in C^1(\mathbb{R}^n)$, we define the gradient $\nabla_gf$ of $f$ in the Riemannian metric $g$ by \begin{equation}\label{2.1} X(f)=\langle \nabla_gf,X\rangle _g, \end{equation} where $X$ is any vector field on $(\mathbb{R}^n,g)$. We denote the Levi-Civita connection of the metric $g$ by $D_g$. For the system \eqref{e1.1}, our main assumption is as follows: \begin{itemize} \item[(A1)] There exist $\varepsilon, \rho_0>0, \Omega_i\subset \Omega $ with smooth boundary $\partial\Omega_i$ and $C^2$ vector fields $H^i$ on $\overline{\Omega_i}, i=1,2,\dots ,m$ such that $\Omega_i\cap \Omega_j =\emptyset, 0\leq i\varepsilon_0\}, \end{gathered} \end{equation} with $\nu^i(x)$ the unit normal of $\partial\Omega_i$ in the Euclidean space $\mathbb{R}^n$, pointing towards the exterior of $\Omega_i$, and $\varepsilon_0 $ is a nonnegative constant satisfying \begin{equation} \Gamma \supset \cup_{i=1}^m\{x\in \partial\Omega_i|00\}. \end{equation} $\Gamma_0$ is widely used to study the control and stabilization of the wave equations with boundary feedbacks. See \cite{ZY20,ZY14,ZY16,Y}. In what follows, we denote by $C$ or $C_i$ any positive constant which may be different from line to line. The following is the stability results of system \eqref{e1.1}. \begin{theorem}\label{thm1.1} Assumption {\rm (A1)} holds, and that $\varepsilon_0$ is sufficiently small and there are positive constants $\lambda>1$ and $T_0>0$ such that \begin{equation} \label{exx1}G(\rho)\geq \lambda G(\rho+T_0)\ \ \ \forall \rho\geq 0. \end{equation} Then there exist constants $C_1, C_2>0$, such that \begin{equation}\label{e2.0} E(t)\le C_1e^{-C_2t}E(0),\quad \forall t> 0. \end{equation} \end{theorem} \begin{example} \label{examp12} \rm Let $G(h)=\int_{h}^{+\infty} e^{-\rho}d\rho=e^{-h},h\geq0$, where $|k(\rho)|= e^{-\rho}$ satisfies \eqref{e2.210}. Since $G'(h)=-e^{-h}<0$, condition \eqref{exx1} holds naturally. \end{example} In this paper, we also consider the stabilization of the problem \begin{equation}\label{ewg.1} \begin{gathered} u_{tt}+\mathscr{A} u+a_1 g_1(u_t(x,t))+a_2g_2(u_t(x,t-\tau(t)))=0\\ (x,t)\in \Omega\times (0,+\infty),\\ u(x,t)\big|_{\Gamma}=0\quad t\in(0,+\infty),\\ u(x,0)=u_0(x),\quad u_t(x,0)=u_1(x)\quad x\in \Omega,\\ u_t(x,t)=h_0(x,t)\quad (x,t)\in \Omega\times (-\tau(0),0), \end{gathered} \end{equation} where $a_1>0$, $a_2\in\mathbb{R}$ are constants and $\tau(t)$ satisfies \begin{equation} \label{ewg.2} \tau(t)\geq 0\quad\text{and} \quad \tau'(t)\leq d <1 \quad \forall t\geq0, \end{equation} where $d$ is a constant. And there exist positive constants $c_1, p\geq1$ such that $g_1, g_2\in C(\mathbb{R})$ satisfy \begin{gather} \label{ewg.3} g_1(0)=0,\quad sg_1(s)\geq \max\{|s|^2, (g_2(s))^2\} \quad\text{for } s\in\mathbb{R}, \\ \label{ewg.4} |g_1(s)|\leq c_1 |s|\quad\text{for } |s|>1. \end{gather} In \cite{ZY10,ZY17,ZY31,ZY5,ZY8,ZY9,ZY16}, the well-posedness and stabilization of the wave equation with a time-varying delay was studied under the assumption that $\tau(t)$ has a upper bound. While in this paper, we will consider the stabilization of system \eqref{ewg.1} with a more general $\tau(t)$, that is, $\tau(t)$ does not need to have a upper bound (See \eqref{ewg.2}). To obtain our stabilization result, we assume that \begin{equation} \frac{|a_2|}{\sqrt{1-d}}0$, $\varphi(t)$ and $\phi(t)$ are strictly increasing functions satisfying \begin{equation} \lim_{t\to+\infty }\varphi(t)=+\infty,\quad \lim_{t\to+\infty }\phi(t)=+\infty. \end{equation} As in \cite{ZY19,ZY18}, we let $h\in C([0,+\infty))$ be a concave increasing function such that \begin{equation} h(0)=0,\quad s^2+(g_1(s))^2\leq h(sg_1(s))\quad\text{for } |s|\leq 1.\label{ewg.8.0} \end{equation} We define \begin{equation} \label{ewg.6.1} F(t)=\sup\{\tau(\rho)+1|0\leq\rho\leq t\} \quad \forall t\geq 0. \end{equation} The following is the stability results of system \eqref{ewg.1}. \begin{theorem} \label{thm1.3} (a) Assume that \begin{equation}\label{ewg.8.1} |g_1(s)|\leq c_2 |s|\quad\text{for } |s|\leq 1. \end{equation} Then there exist constants $C_1, C_2 >0$, such that \begin{equation} \label{ewg.8.2} J(t)\le \frac{C_1 J(0)}{t^{C_2}},\quad \forall t> 0. \end{equation} (b) Assume that \begin{equation}\label{ewg.8.3} \lim_{t\to+\infty }\frac{F(t)}{t}=0. \end{equation} Then \begin{equation}\label{ewg.8.3.1} \lim_{t\to+\infty }\frac{F(\phi(t))}{t}=0. \end{equation} Also there exist constants $C_1,C_2 >0$, such that \begin{equation} \label{ewg.8.4} J(t)\leq C_1 h\Big(\frac{C_2}{\phi(t)}J(0)\Big) +C_1 \frac{F(\phi(t))}{t}J(0),\forall t> 0. \end{equation} \end{theorem} \begin{example} \label{examp1.4} \rm Let $\tau(t)=\frac{d^2}{t+1}$, then $\tau'(t)=-\frac{d^2}{(t+1)^2}$, which implies relation \eqref{ewg.2}. Since \begin{equation}\label{} \lim_{t\to+\infty }\frac{F(t)}{t} =\lim_{t\to+\infty }\frac{sup\{d^2/(\rho+1)+1|0\leq\rho\leq t\}}{t}=0, \end{equation} condition \eqref{ewg.8.3} is satisfied. \end{example} This article is organized as follows. Section 2 is devoted to presenting the well-posedness of systems \eqref{e1.1} and \eqref{ewg.1}. The technical details of the proof for Theorems \ref{thm1.1} and \ref{thm1.3} are given in Section 3 and Section 4, respectively. \section{Well-posedness} To obtain the well-posedness of system \eqref{e1.1}, we define \begin{gather} L_a^2(\Omega\times(-\infty,t))=\Big\{ u: \int_0^\infty \int_{\Omega}a(x)G(\rho)u^2(x,t-\rho)\,dx\,d\rho <+\infty \Big\}.\label{E001}\\ \begin{aligned} &L_a^2(\Omega, H^1(-\infty,t))\\ &=\Big\{ u: \int_0^\infty \int_{\Omega}a(x)G(\rho)(u^2(x,t-\rho) +u^2_\rho(x,t-\rho))\,dx\,d\rho<+\infty \Big\}.\label{EE001} \end{aligned} \end{gather} Note that system \eqref{e1.1} is a linear equation and the kernel $k(\cdot)$ defined on $[0,+\infty)$ does not change with time $t$. Using the methods in \cite{NP}, by a similar proof, we obtain the following well-posedness result. \begin{theorem}\label{thm1.2} For any initial datum $(u_0,u_1, f_0)\in H^1_{0}(\Omega)\times L^2(\Omega) \times L_a^2(\Omega\times(-\infty,0))$, there exists a unique solution $u$ of system \eqref{e1.1} satisfying $$ u\in C^1([0,+\infty),L^2(\Omega))\cap C([0,+\infty),H^1_{0}(\Omega)),\quad u_t\in C([0,+\infty), L_a^2(\Omega\times(-\infty,t)). $$ Moreover, if $(u_0,u_1, f_0)\in (H^2(\Omega)\cap H^1_{0}(\Omega)) \times H^1_{0}(\Omega)\times L_a^2(\Omega, H^1(-\infty,0))$ satisfies the compatibility condition $f_0(\cdot,0)=u_1$, then the unique solution $u$ satisfies \[ u\in C^1([0,+\infty),H^1_{0}(\Omega))\cap C([0,+\infty),H^2(\Omega) \cap H^1_{0}(\Omega)) \] and $u_t\in C([0,+\infty),L_a^2(\Omega, H^1(-\infty,t) )$. \end{theorem} To obtain the stabilization of system \eqref{ewg.1}, we assume system \eqref{ewg.1} is well-posed such that \begin{equation} \int_{0}^{\phi(0)} \int_{\Omega}h_{0}(x,\varphi(t))g_1(h_{0}(x,\varphi(t)))\,dx\,dt <+\infty \end{equation} and $ u\in C^1([0,+\infty),L^2(\Omega))\cap C([0,+\infty), H^1_{\Gamma_2}(\Omega))$. \section{Proofs of Theorem \ref{thm1.1}} The following lemma is given in \cite[Lemma 2.1]{Y} to introduce the relations between the standard dot metric $\langle \cdot,\cdot\rangle $ and the Riemannian metric $g=\langle \cdot,\cdot\rangle _g$. \begin{lemma}\label{lem1} Let $x=(x_1,\cdots,x_n)$ be the natural coordinate system in $\mathbb{R}^n$. Let $f$, $h$ be functions and let $H$, $X$ be vector fields. Then \begin{itemize} \item[(a)] \begin{equation}\label{2.2} \langle H(x),A(x)X(x)\rangle _g=\langle H(x),X(x)\rangle ,\quad x\in\mathbb{R}^n; \end{equation} \item[(b)] \begin{equation}\label{2.3} \nabla_gf=\sum_{i=1}^n\Big(\sum_{j=1}^na_{ij}(x)f_{x_j}\Big)\frac{\partial}{\partial x_i}=A(x)\nabla f,\quad x\in\mathbb{R}^n, \end{equation} where $\nabla f$ is the gradient of $f$ in the standard metric; \item[(c)] \begin{equation}\label{2.4} \nabla_gf(h)=\langle \nabla_gf, \nabla_gh\rangle _g=\langle \nabla f,\,\,A(x)\nabla h\rangle , \quad x\in\mathbb{R}^n, \end{equation} where the matrix $A(x)$ is given in the formula \eqref{mu6}. \end{itemize} \end{lemma} Let \begin{equation} E_{0}(t)=\frac12\int_{\Omega}\big(u_t^2+|\nabla_gu|_g^2\big)dx\,.\label{E0} \end{equation} Using \eqref{e2.1} and \eqref{ewg.7}, we have \begin{gather} E(t)=E_{0}(t)+\xi\int_0^\infty \int_{\Omega}a(x)G(\rho)u^2_t(x,t-\rho)\,dx\,d\rho, \label{E1}\\ J(t)=E_{0}(t)+\eta\int_{t}^{\phi(t)} \int_{\Omega}u_t(x,\varphi(\rho)) g_1(u_t(x,\varphi(\rho)))\,dx\,d\rho.\label{E1.1} \end{gather} Let $\hat{\Omega}$ be a subset of $\Omega$, we define \begin{gather} E_{0}(\hat{\Omega},t)=\frac12\int_{\hat{\Omega}} \big(u_t^2+|\nabla_gu|_g^2\big)dx,\\ E(\hat{\Omega},t)=E_{0}(\hat{\Omega},t) +\xi\int_0^\infty \int_{\hat{\Omega}}a(x)G(\rho)u^2_t(x,t-\rho)\,dx\,d\rho. \end{gather} \begin{lemma}\label{lem2.2} Suppose that $\eqref{mu2}$ holds. Let $u(x,t)$ be the solution of \eqref{e1.1}. Then there exist constants $C_1,C_2>0$ such that \begin{gather} \label{e2.2} E(0)-E(T)\ge C_1\int_0^T \int_{\Omega}a(x)\Big(u_t^2(x,t) +\int_{0}^{\infty}|k(\rho)|u^2_t(x,t-\rho) d\rho\Big)\,dx\,dt, \\ \label{n1.1} E(0)-E(T)\leq C_2\int_0^T \int_{\Omega}a(x)\Big(u_t^2(x,t) +\int_{0}^{\infty} |k(\rho)|u^2_t(x,t-\rho) d\rho\Big) \,dx\,dt, \end{gather} where $T> 0$. The assertion \eqref{e2.2} implies that $E(t)$ is decreasing. \end{lemma} \begin{proof} Differentiating \eqref{e2.1}, we obtain \begin{equation} \begin{aligned} E'(t)&=\int_{\Omega}\left(u_tu_{tt}+\nabla_g u\cdot\nabla u_t\right)dx \\ &\quad +2\xi\int_0^{+\infty}\int_{\Omega}a(x)G(\rho)u_{tt}(x,t-\rho) u_t(x,t-\rho) \,dx\,d\rho.\label{2.611} \end{aligned} \end{equation} Applying Green's formula, the fact that $$ u_t(x,t-\rho)=-u_{\rho}(x,t-\rho),\quad u_{tt}(x,t-\rho) =u_{\rho\rho}(x,t-\rho), $$ and integrating by parts, we obtain \begin{equation} \label{2.6} \begin{aligned} E'(t)&=\int_{\Omega}a(x)\Big[\Big(-\mu_1u_t^2(x,t)-\mu_2u_t(x,t) \int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho\Big) \\ &\quad+\xi\Big(u_t^2(x,t)-\int_{0}^{\infty}|k(\rho)|u^2_t(x,t-\rho) d\rho\Big)\Big]dx. \end{aligned} \end{equation} With \eqref{e2.210} we deduce that \begin{align*} &\Big|u_t(x,t)\int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho\Big| \\ &\leq \frac 12u^2_t(x,t)+\frac12\Big(\int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho\Big)^2 \\ &\leq \frac 12u^2_t(x,t)+\frac 12\int_0^{\infty}|k(\rho)|d\rho \int_0^{\infty}|k(\rho)|u^2_t(x,t-\rho)d\rho \\ &=\frac12u^2_t(x,t)+\frac12 \int_0^{\infty}|k(\rho)|u^2_t(x,t-\rho)d\rho. \end{align*} Then, it follows from \eqref{mu2} and \eqref{2.6} that \begin{align} E'(t)\leq -C_1\int_{\Omega}a(x) \Big(u_t^2(x,t)+\int_{0}^{\infty}|k(\rho)|u^2_t(x,t-\rho) d\rho\Big)dx,\label{2.7} \\ E'(t)\geq -C_2\int_{\Omega}a(x)\Big(u_t^2(x,t) +\int_{0}^{\infty}|k(\rho)|u^2_t(x,t-\rho) d\rho\Big)dx,\label{n1.2} \end{align} where $C_1,C_2> 0$ and $C_1$ satisfies $$ C_1=\min\big\{\xi-\frac{\mu_2}2,\; \mu_1-\frac{\mu_2}2-\xi\big\}. $$ Then inequalities \eqref{e2.2} and \eqref{n1.1} follow directly from \eqref{2.7} and \eqref{n1.2} by integrating from $0$ to $T$. \end{proof} By a similar proof as in \cite[Proposition 2.1]{Y}, we have the following identities. \begin{lemma}\label{lemc.1} Suppose that $u(x,t)$ solves the equation \begin{equation} \label{e2.x4} u_{tt}+\mathscr{A} u+a(x)\Big[\mu_1u_t(x,t)+\mu_2\int_{0}^{\infty}k(\rho)u_t(x,t-\rho) d\rho\Big]=0 \end{equation} for $ (x,t)\in \Omega\times (0,+\infty)$, and that $\mathcal {H}$ is a vector field defined on $\overline \Omega$. Then \begin{equation} \label{e2.4} \begin{aligned} &\int_0^T\int_{\Gamma}\frac{\partial u}{\partial\nu_{\mathscr{A}}}\mathcal {H}(u) d\Gamma dt+\frac12\int_0^T\int_{\Gamma} \big(u_t^2-|\nabla_g u|_g^2\big)\mathcal {H}\cdot\nu d\Gamma dt \\ &=(u_t,\mathcal {H}(u))\big|^T_0 +\int_0^T\int_{\Omega}a(x)\mathcal {H}(u)[\mu_1u_t(x,t) +\mu_2 \int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho]\,dx\,dt \\ &\quad +\int_0^T\int_{\Omega}D_g\mathcal {H}(\nabla_g u,\nabla_g u) \,dx\,dt +\frac12\int_0^T\int_{\Omega}\left(u_t^2-\left |\nabla_g u\right|_g^2\right)\operatorname{div}\mathcal {H} \,dx\,dt . \end{aligned} \end{equation} Moreover, assuming that $P\in C^1(\overline \Omega)$, we have \begin{equation} \label{e2.5} \begin{aligned} &\int_0^T\int_{\Omega}\big(u_t^2-|\nabla_g u|_g^2\big)P \,dx\,dt \\ &= (u_t,u P)\big|^T_0+\frac12\int_0^T \int_{\Omega}\nabla_gP(u^2) \,dx\,dt-\int_0^T\int_{\Gamma}Pu\frac{\partial u}{\partial\nu_{\mathscr{A}}}d\Gamma dt \\ &\quad +\int_0^T\int_{\Omega}a(x)Pu[\mu_1u_t(x,t) +\mu_2 \int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho]\,dx\,dt. \end{aligned} \end{equation} \end{lemma} The following is the energy inequality for our stabilization problem. \begin{lemma}\label{lem2.3} Let the conditions in Theorem \ref{thm1.1} hold, and let $u(x,t)$ be the solution of system \eqref{e1.1}. Then there exists $\overline{T}>0$ such that, for $T>\overline{T}$, there exists a positive constant $C_T$ such that \begin{equation} \label{e2.7} E(0)\le C_T\int_0^T\int_{\Omega}a(x)\Big[u_t^2(x,t) +\int_{0}^{\infty}|k(\rho)|u^2_t(x,t-\rho) d\rho\Big]\,dx\,dt. \end{equation} \end{lemma} \begin{proof} Let $\phi^i\in C_0^\infty(\mathbb{R}^n)$ satisfy $0\leq\phi^i\leq 1$ and \begin{equation} \label{xx.4} \phi^i=\begin{cases} 1, & x\in \Omega_i\backslash V_1\\ 0, & x\in V_2, \end{cases} \end{equation} for $1\leq i\leq m$. Set $$ \mathcal {H}=\phi^iH^i,\quad P=\frac12\big(\operatorname{div}(\phi^iH^i) -\rho_0\big), \quad \Omega_i=\Omega. $$ Substituting \eqref{e2.5} into \eqref{e2.4}, we have \begin{equation} \label{xx.5} \begin{aligned} &\Pi_{\partial\Omega_i} \\ &=\big(u_t,\phi^iH^i(u)+Pu\big)\big|_0^T +\frac{\rho_0}{2}\int_0^T E_0(\Omega_i,t) dt \\ &\quad +\int_0^T \int_{\Omega_i}\Big(D_g(\phi^iH^i)(\nabla_g u,\nabla_g u) -\rho_0|\nabla_g u|^2_g+\frac12\nabla_gP(u^2)\Big) \,dx\,dt \\ &\quad +\int_0^T\int_{\Omega_i}a(x)\phi^iH^i(u)[\mu_1u_t(x,t) +\mu_2 \int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho]\,dx\,dt \\ &\quad +\int_0^T\int_{\Omega_i}a(x)Pu[\mu_1u_t(x,t) +\mu_2 \int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho]\,dx\,dt, \end{aligned} \end{equation} where \begin{equation} \label{xx.6} \begin{aligned} \Pi_{\partial\Omega_i} &=\int_0^T\int_{\partial\Omega_i}\frac{\partial u}{\partial\nu_{\mathscr{A}}} \left(\phi^iH^i(u) +uP\right) d\Gamma dt\\ &\quad +\frac12\int_0^T\int_{\partial\Omega_i}\left(u_t^2-|\nabla_g u|_{g}^2\right)\phi^iH^i\cdot\nu d\Gamma dt. \end{aligned} \end{equation} Note that $ \partial\Omega_i \in V_2 \cup \Gamma^i_1$, where $\Gamma^i_1= \{x\in \partial\Omega_i \cap \Gamma | H^i(x)\cdot\nu(x)\leq 0\}$. We decompose $\Pi_{\partial\Omega_i}$ as \begin{equation} \label{xx.7} \Pi_{\partial\Omega_i}=\Pi_{\partial\Omega_i \cap V_2} +\Pi_{(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2}, \end{equation} where \begin{gather} \label{xx.8} \begin{aligned} \Pi_{\partial\Omega_i \cap V_2} &=\int_0^T\int_{\partial\Omega_i \cap V_2}\frac{\partial u}{\partial \nu_{\mathscr{A}}}\left(\phi^iH^i(u) +uP\right) d\Gamma dt \\ &\quad +\frac12\int_0^T\int_{\partial\Omega_i \cap V_2}\left(u_t^2-|\nabla_g u|_{g}^2\right)\phi^iH^i\cdot\nu d\Gamma dt, \end{aligned} \\ \label{xx.9} \begin{aligned} \Pi_{(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2} &= \int_0^T\int_{(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2} \frac{\partial u}{\partial\nu_{\mathscr{A}}}\left(\phi^iH^i(u) +uP\right) d\Gamma dt \\ &\quad +\frac12\int_0^T\int_{(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2} \left(u_t^2-|\nabla_g u|_{g}^2\right)\phi^iH^i\cdot\nu d\Gamma dt. \end{aligned} \end{gather} From \eqref{xx.4}, we have \begin{equation}\label{xx.10} \Pi_{\partial\Omega_i \cap V_2}=0. \end{equation} Since $u\big|_{(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2}=0$, we obtain $\nabla_{\Gamma_g }u\big|_{(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2}=0$; that is, \begin{equation} \nabla_g u=\frac{\partial u}{\partial\nu_{\mathscr{A}}} \frac{\nu_{\mathscr{A}}}{|\nu_{\mathscr{A}}|_g^2}\quad\text{for}\quad x\in(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2.\label{xx.11} \end{equation} Similarly, we have \begin{equation} H(u)=\langle H,\nabla_g u\rangle _g =\frac{\partial u}{\partial\nu_{\mathscr{A}}} \frac{H\cdot\nu}{|\nu_{\mathscr{A}}|_g^2}\quad\text{for } x\in(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2.\label{xx.12} \end{equation} Using formulas \eqref{xx.11} and \eqref{xx.12} in \eqref{xx.9}, with \eqref{exx.2} and \eqref{xx.3}, we obtain \begin{equation}\label{xx.13} \begin{aligned} \Pi_{(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2} &=\frac12\int_0^T \int_{(\partial\Omega_i \cap \Gamma^i_1)\backslash V_2} \Big(\frac{\partial u}{\partial\nu_{\mathscr{A}}}\Big)^2 \frac{H\cdot\nu}{|\nu_{\mathscr{A}}|_g^2} d\Gamma dt \\ &\leq C\varepsilon_0\int_0^T \int_{\Gamma}\Big(\frac{\partial u}{\partial\nu_{\mathscr{A}}}\Big)^2 d\Gamma dt. \end{aligned} \end{equation} Let $H_1$ be a $C^1$ vector field on $\overline{\Omega}$ satisfying \begin{equation}\label{xx.13_1} H_1=\frac{\nu_{\mathscr{A}}}{|\nu_{\mathscr{A}}|_g^2}\quad x\in \Gamma. \end{equation} Replacing $\mathcal {H} $ with $H_1$ in \eqref{e2.4} and noting that $u\big|_{\Gamma}=0$, by a similar discussion on $\Gamma$ with \eqref{xx.11} $\sim$ \eqref{xx.13} we have \begin{equation} \label{xx.13_2} \begin{aligned} &\frac12\int_0^T \int_{\Gamma}\Big(\frac{\partial u}{\partial\nu_{\mathscr{A}}}\Big)^2 d\Gamma dt \\ &=(u_t,H_1(u))\big|^T_0+\int_0^T dt\int_{\Omega}D_g H_1(\nabla_g u,\nabla_g u) dx \\ &\quad +\frac12\int_0^T dt\int_{\Omega}\big(u_t^2-|\nabla_g u|_g^2\big) \operatorname{div} H_1 dx \\ &\quad +\int_0^T dt\int_{\Omega}a(x) H_1(u)\Big[\mu_1u_t(x,t) +\mu_2 \int_0^{\infty}k(\rho) u_t(x,t-\rho)d\rho\Big]dx \\ &\leq C_1(E(0)+E(T))+C_2\int_0^T E_0(t) dt \\ &\quad +C_3\int_0^T\int_{\Omega}a(x)\big[u_t^2(x,t) + \int_0^{\infty}|k(\rho)| u^2_t(x,t-\rho)d\rho\big]\,dx\,dt. \end{aligned} \end{equation} Substituting \eqref{xx.10}, \eqref{xx.13} and \eqref{xx.13_2} into \eqref{xx.5}, and using \eqref{mu3} and \eqref{exx.1}, we obtain \begin{equation} \label{xx.14} \begin{aligned} &\int_0^T E_0(\Omega_i \backslash V_1,t)dt \\ &\leq C_4(E(0)+E(T))+C_5\varepsilon_0\int_0^T E_0(t)dt +\int_0^T\int_{\Omega_i}(C_\alpha u^2+\alpha|\nabla_g u|_g^2)\,dx\,dt \\ &\quad +C_6\int_0^T\int_{\Omega}a(x)[u_t^2(x,t) +\int_{0}^{\infty} |k(\rho)|u^2_t(x,t-\rho) d\rho+|\nabla_g u|_g^2]\,dx\,dt, \end{aligned} \end{equation} where $\alpha$ is sufficiently small. Then, noting that $\Omega\subset(\bigcup_{i=1}^m \Omega_i\cup V_1)$, we have \begin{equation} \label{xx.15} \begin{aligned} &\int_0^T E_0(\Omega \backslash V_1,t)dt \\ &\leq C_4m(E(0)+E(T))+C_5m\varepsilon_0\int_0^T E_0(t)dt\\ &\quad +\sum_{i=1}^m\int_0^T\int_{\Omega_i}(C_\alpha u^2+\alpha|\nabla_g u|_g^2) \,dx\,dt +C_6m\int_0^T\int_{\Omega}a(x)[u_t^2(x,t) \\ &\quad +\int_{0}^{\infty} |k(\rho)|u^2_t(x,t-\rho) d\rho+|\nabla_g u|_g^2]\,dx\,dt \\ &\leq C_4m(E(0)+E(T))+C_5m\varepsilon_0\int_0^T E_0(t)dt +C_7\int_0^T\int_{\Omega}u^2\,dx\,dt \\ &\quad +C_8\int_0^T\int_{\Omega}a(x)[u_t^2(x,t) +\int_{0}^{\infty} |k(\rho)|u^2_t(x,t-\rho) d\rho+|\nabla_g u|_g^2]\,dx\,dt. \end{aligned} \end{equation} Then, using \eqref{exx.1}, we have \begin{equation} \label{xx.16} \begin{aligned} &\int_0^T E_0(t)dt\\ &\leq C_9(E(0)+E(T))+C_{10}\int_0^T\int_{\Omega}u^2\,dx\,dt \\ &\quad +C_{11}\int_0^T\int_{\Omega}a(x)[u_t^2(x,t) +\int_{0}^{\infty} |k(\rho)|u^2_t(x,t-\rho) d\rho+|\nabla_g u|_g^2]\,dx\,dt. \end{aligned} \end{equation} Set $P=a(x)$ and substituting identity \eqref{e2.5} into identity \eqref{xx.16}, we obtain \begin{equation} \label{ed.x72} \begin{aligned} \int_0^T E_0(t)dt &\leq C_{12}(E(0)+E(T))+C_{13}\int_0^T\int_{\Omega}u^2\,dx\,dt \\ &\quad +C_{14}\int_0^T\int_{\Omega}a(x)[u_t^2(x,t) + \int_0^{\infty}|k(\rho)| u^2_t(x,t-\rho)d\rho]\,dx\,dt. \end{aligned} \end{equation} From \eqref{exx1}, we have \begin{equation} \label{ed.x81} \begin{aligned} &\int_0^{T_0}\int_{0}^{\infty}\int_{\Omega}a(x) |k(\rho)|u^2_t(x,t-\rho) \,dx\,d\rho\,dt \\ &=\int_0^{T_0}\int_{-t}^{\infty}\int_{\Omega}a(x)|k(t+\rho) |u^2_t(x,-\rho)\,dx\,d\rho\,dt \\ &\geq \int_{0}^{\infty}\int_{\Omega}a(x) \int_0^{T_0} |k(t+\rho)|u^2_t(x,-\rho)\,dx\,d\rho\,dt \\ &\geq \big(1-\frac{1}{\lambda}\big) \int_{0}^{\infty}\int_{\Omega}a(x) G(\rho)u^2_t(x,-\rho)\,dx\,d\rho\,dt. \end{aligned} \end{equation} Then, for $T\geq T_0$, with \eqref{E1} and \eqref{e2.2} we obtain \begin{equation} \label{ed.x8} \begin{aligned} &C_{12}(E(T)+E(0))+E(0)\leq (2C_{12}+1)E(0) \\ &=(2C_{12}+1)E_0(0)+(2C_{12}+1)\xi\int_{0}^{\infty}\int_{\Omega}a(x) G(\rho)u^2_t(x,t-\rho) \,dx\,d\rho\,dt \\ &\leq (2C_{12}+1)E_0(0) \\ &\quad +(2C_{12}+1)\big(\frac{\lambda}{\lambda-1}\big) \xi\int_0^{T}\int_{0}^{\infty}\int_{\Omega}a(x) |k(\rho)|u^2_t(x,t-\rho) \,dx\,d\rho\,dt. \end{aligned} \end{equation} Note that for $T\geq 2C_{12}+1$, \begin{equation} \label{y20131031.1} \begin{aligned} (2C_{12}+1)E_0(0) &\leq \int_0^T E_0(t)dt+ \int_0^{2C_{12}+1}( E_0(0)-E_0(t))dt \\ &=- \int_0^{2C_{12}+1}\int_0^t \int_{\Omega}a(x)u_t(x,t')[\mu_1u_t(x,t')\\ &\quad +\mu_2 \int_0^{\infty}k(\rho) u_t(x,t'-\rho)d\rho]\,dx\,dt'dt +\int_0^T E_0(t)dt \\ &\leq (2C_{12}+1)(\mu_1+\frac{\mu_2}{2})\int_0^{2C_{12}+1} \int_{\Omega}a(x)[u_t^2(x,t) \\ &\quad +\int_{0}^{\infty} |k(\rho)|u^2_t(x,t-\rho) d\rho]\,dx\,dt +\int_0^T E_0(t)dt. \end{aligned} \end{equation} Substituting \eqref{ed.x8}-\eqref{y20131031.1} into \eqref{ed.x72}, for sufficiently large $T$, we obtain \begin{equation} \label{ed.x9} \begin{aligned} E(0)&\leq C_{15}\int_0^{T}\int_{\Omega}a(x) \Big(u_t^2(x,t)+\int_{0}^{\infty} |k(\rho)|u^2_t(x,t-\rho) d\rho\Big)\,dx\,dt \\ &\quad +C_{13}\int_0^{T} \int_{\Omega}u^2\,dx\,dt. \end{aligned} \end{equation} Estimate \eqref{e2.7} follows from the inequality \eqref{ed.x9} by a compactness-uniqueness argument as in \cite{ZY1}. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] Let $\overline{T}>0$ be given by Lemma \ref{lem2.3}. Then it follows from \eqref{e2.2} and \eqref{e2.7} that, for $T>\overline{T}$, \begin{equation} \label{re} \begin{aligned} E(0) &\leq C_T\int_0^T\int_{\Omega}a(x)(u_t^2(x,t)+\int_{0}^{\infty} |k(\rho)|u^2_t(x,t-\rho) d\rho)\,dx\,dt \\ &\leq C_TC_1^{-1}(E(0)-E(T)). \end{aligned} \end{equation} Then \begin{equation} E(T) \le \frac{C_TC_1^{-1}-1}{C_TC_1^{-1}}E(0).\label{E4} \end{equation} Estimate \eqref{e2.0} follows from the inequality \eqref{E4}. \end{proof} \section{Proof of Theorem \ref{thm1.3}} \begin{lemma} \label{lem4.1} Suppose that \eqref{ewg.5} holds, and let $u(x,t)$ be the solution of \eqref{ewg.1}. Then there exist constants $C_1,C_2>0$ such that \begin{gather}\label{ewg.9} \begin{aligned} &J(T_1)-J(T_2) \\ &\ge C_1\int_{T_1}^{T_2} \int_{\Omega}(u_t(x,t)g_1(u_t(x,t))+u_t(x,\varphi(t))g_1(u_t(x,\varphi(t))) \,dx\,dt, \end{aligned} \\ \label{ewg.9.1} \begin{aligned} &J(T_1)-J(T_2) \\ &\leq C_2\int_{T_1}^{T_2} \int_{\Omega}(u_t(x,t)g_1(u_t(x,t)) +u_t(x,\varphi(t))g_1(u_t(x,\varphi(t)))\,dx\,dt, \end{aligned} \end{gather} where $T_2>T_1\ge 0$. Assertion \eqref{ewg.9} implies that $J(t)$ is decreasing. \end{lemma} \begin{proof} Differentiating \eqref{ewg.7}, with \eqref{ewg.6}, we obtain \begin{equation} \label{ewg.10} \begin{aligned} J'(t)&=\int_{\Omega}\left(u_tu_{tt}+\nabla_g u\cdot\nabla u_t\right)dx +\eta\phi'(t)\int_{\Omega} u_t(x,t)g_1(u_t(x,t))dx \\ &\quad -\eta\int_{\Omega} u_t(x,\varphi(t))g_1(u_t(x,\varphi(t)))dx . \end{aligned} \end{equation} Note that \begin{equation} \phi'(t)=\frac{1}{\varphi'(\phi(t))}=\frac{1}{1-\tau'(\phi(t))}\leq \frac{1}{1-d}, \end{equation} by Green's formula, we deduce that \begin{equation} \label{ewg.11} \begin{aligned} J'(t)&=\eta\phi'(t)\int_{\Omega} u_t(x,t)g_1(u_t(x,t))dx \\ &\quad -\eta\int_{\Omega} u_t(x,\varphi(t))g_1(u_t(x,\varphi(t)))dx \\ &\quad +\int_{\Omega}[-a_1 u_tg_1(u_t)-a_2 u_tg_2( u_t(x,\varphi(t)))]dx \\ &\leq \int_{\Omega}\Big[-\eta u_t(x,\varphi(t))g_1(u_t(x,\varphi(t))) +\frac{\sqrt{1-d}|a_2|}{2}g^2_2( u_t(x,\varphi(t)))\Big]dx \\ &\quad +\int_{\Omega}\Big(-a_1 u_tg_1(u_t)+\frac{|a_2|}{2\sqrt{1-d}}u^2_t + \frac{\eta}{1-d} u_tg_1(u_t) \Big)dx\,. \end{aligned} \end{equation} From \eqref{ewg.3}, \eqref{ewg.5} and \eqref{ewg.8}, we obtain \begin{equation} J'(t)\leq -C_1\int_{\Omega}[u_t(x,t)g_1(u_t(x,t)) +u_t(x,\varphi(t))g_1(u_t(x,\varphi(t)))]dx\label{ewg.12}, \end{equation} where $C_1>0$ satisfies $$ C_1=\min\big\{a_1-\frac{|a_2|}{2\sqrt{1-d}}-\frac{\eta}{1-d},\ \ \eta-\frac{\sqrt{1-d}|a_2|}{2}\big\}. $$ Note that \begin{equation} \phi'(t)\ge 0 \quad \forall t\ge0. \end{equation} From the first step of \eqref{ewg.11}, with \eqref{ewg.3} we obtain \begin{equation} J'(t)\geq -C_2\int_{\Omega}[u_t(x,t)g_1(u_t(x,t)) +u_t(x,\varphi(t))g_1(u_t(x,\varphi(t)))]dx\label{ewg.12.1}, \end{equation} where $C_2$ is a positive constant. Then the inequality \eqref{ewg.9}/\eqref{ewg.9.1} follows directly from \eqref{ewg.12}/\eqref{ewg.12.1} integrating from $T_1$ to $T_2$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.3}] Let $T_2>T_1\ge 0$. Multiplying \eqref{ewg.1} by $u$ and integrating from $T_1$ to $T_2$, we have \begin{equation} \label{ewg.13} \begin{aligned} &\int_{T_1}^{T_2}\int_{\Omega}\left(u_t^2-\left |\nabla_g u\right|_g^2\right) \,dx\,dt \\ &= (u_t,u )\big|_{T_1}^{T_2}+\int_{T_1}^{T_2} \int_{\Omega}u\left(a_1 g_1(u_t(x,t))+a_2g_2(u_t(x,t-\tau(t)))\right) \,dx\,dt. \end{aligned} \end{equation} Then \begin{equation} \label{ewg.13.1} \begin{aligned} &\int_{T_1}^{T_2}E_0(t)dt\\ &=2\int_{T_1}^{T_2}\int_{\Omega}u_t^2\,dx\,dt -\int_{T_1}^{T_2}\int_{\Omega}\big(u_t^2-|\nabla_g u|_g^2\big) \,dx\,dt \\ &=2\int_{T_1}^{T_2}\int_{\Omega}u_t^2\,dx\,dt \\ &\quad -(u_t,u )\big|_{T_1}^{T_2}-\int_{T_1}^{T_2}\int_{\Omega}u \left(a_1g_1(u_t)+a_2g_2(u_t(\varphi(t)))\right) \,dx\,dt \\ &\leq 2\int_{T_1}^{T_2}\int_{\Omega}u_t^2dx +C(J(T_1)+J(T_2)) \\ &\quad +\varepsilon \int_{T_1}^{T_2}\int_{\Omega}u^2\,dx\,dt +C_\varepsilon \int_{T_1}^{T_2}\int_{\Omega}[g^2_1(u_t(x,t)) +g^2_2(u_t(x,\varphi(t)))]\,dx\,dt. \end{aligned} \end{equation} From \eqref{ewg.3}, we have \begin{equation} \label{ewg.13.2} g_1(s)\ge |s|\quad for \quad s\in\mathbb{R}. \end{equation} Then, from \eqref{ewg.13.1} it follows that \begin{equation}\label{ewg.14} \begin{aligned} \int_{T_1}^{T_2}E_0(t)dt &\leq \widetilde{C}(J(T_1)+J(T_2)) +C \int_{T_1}^{T_2}\int_{\Omega}\Big[g^2_1(u_t(x,t)) \\ &\quad +u_t(x,\varphi(t))g_1(u_t(x,\varphi(t)))\Big]\,dx\,dt, \end{aligned} \end{equation} where $\widetilde{C}$ is a positive constant. \smallskip \noindent\textbf{Proof of (a).} From \eqref{ewg.4} and \eqref{ewg.8.1} we have \begin{equation} \label{ewg.14.1} g^2_1(s)\leq \max\{c_1,c_2\} sg_1(s)\quad \text{for } s\in\mathbb{R}. \end{equation} Then, from \eqref{ewg.14} it follows that \begin{equation} \label{ewg.14.2} \begin{aligned} \int_{T_1}^{T_2}E_0(t)dt &\leq \widetilde{C}(J(T_1)+J(T_2)) +C \int_{T_1}^{T_2}\int_{\Omega}\Big[u_t(x,t)g_1(u_t(x,t))\\ &\quad +u_t(x,\varphi(t)) g_1(u_t(x,\varphi(t)))\Big]\,dx\,dt. \end{aligned} \end{equation} Since $J(t)$ is decreasing, from \eqref{E1.1} we deduce that \begin{equation} \label{ewg.15} \begin{aligned} &\widetilde{C}(J(T_1)+J(T_2))+J(T_1)\\ &\leq (2\widetilde{C}+1)J(T_1)\\ &=(2\widetilde{C}+1)E_0(T_1) +(2\widetilde{C}+1)\eta\int_{T_1}^{\phi(T_1)} \int_{\Omega}u_t(x,\varphi(t))g_1(u_t(x,\varphi(t)))d\Gamma dt. \end{aligned} \end{equation} From \eqref{ewg.3}, we deduce that \begin{equation} \label{ewg.16} \begin{aligned} &(2\widetilde{C}+1)E_0(T_1)\\ &= \int_{T_1}^{T_1+2\widetilde{C}+1} E_0(t)dt + \int_{T_1}^{T_1+2\widetilde{C}+1}( E_0(T_1)-E_0(t))dt \\ &= \int_{T_1}^{T_1+2\widetilde{C}+1} E_0(t)dt +\int_{T_1}^{T_1+2\widetilde{C}+1}\int_{0}^{t} \int_{\Omega}u_t(x,t')\big[a_1 g_1(u_t(x,t')) \\ &\quad +a_2g_2(u_t(x,\varphi(t')))\big]]\,dx\,dt'dt \\ &\leq \int_{T_1}^{T_1+2\widetilde{C}+1} E_0(t)dt + (2\widetilde{C}+1)\big(a_1+\frac{|a_2|}{2}\big) \\ &\quad\times \int_{T_1}^{T_1+2\widetilde{C}+1}\int_{\Omega}[u_t(x,t)g_1(u_{t}(x,t)) +g_2^2(u_t(x,\varphi(t)))]\,dx\,dt \\ &\leq \int_{T_1}^{T_1+2\widetilde{C}+1} E_0(t)dt + (2\widetilde{C}+1)\big(a_1+\frac{|a_2|}{2}\big) \int_{T_1}^{T_1+2\widetilde{C}+1}\int_{\Omega}[u_t(x,t)g_1(u_{t}(x,t)) \\ &\quad +u_t(x,\varphi(t))g_1(u_t(x,\varphi(t)))]\,dx\,dt. \end{aligned} \end{equation} Substituting \eqref{ewg.15} and \eqref{ewg.16} into \eqref{ewg.14.2}, for $T_2\ge\max\{T_1 +2\widetilde{C}+1,\phi(T_1)\}$, with \eqref{ewg.9} we have \begin{equation} \label{ewg.17} \begin{aligned} J(T_1) &\leq C \int_{T_1}^{T_2}\int_{\Omega}[u_t(x,t)g_1(u_{t}(x,t))+u_t(x,\varphi(t)) g_1(u_t(x,\varphi(t)))]\,dx\,dt \\ &\leq CC_1^{-1}(J(T_1)-J(T_2)). \end{aligned} \end{equation} Then \begin{equation}\label{ewg.18} J(T_2)\leq \lambda J(T_1), \end{equation} where $0<\lambda<1$ is a constant and $T_2\ge\max\{T_1 +2\widetilde{C}+1,\phi(T_1)\}$. From \eqref{ewg.6}, we have \begin{equation}\label{ewg.19} \phi'(t)=\frac{1}{\varphi'(\phi(t))}=\frac{1}{1-\tau'(\phi(t))} \leq \frac{1}{1-d},\quad \forall t\ge -\tau(0). \end{equation} Then \begin{equation} \label{ewg.20} \phi(t)\leq \frac{t+\tau(0)}{1-d}-\tau(0)\quad \forall t\ge-\tau(0). \end{equation} Let $d