\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 12, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/12\hfil Blow-up and general decay of solutions] {Blow-up and general decay of solutions for a nonlinear viscoelastic equation} \author[W. Chen, Y. Xiong \hfil EJDE-2013/12\hfilneg] {Wenying Chen, Yangping Xiong} % in alphabetical order \address{Wenying Chen \newline College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing 404000, China} \email{wenyingchenmath@yahoo.com} \address{Yangping Xiong \newline Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China} \email{xiongyangping@gmail.com} \thanks{Submitted November 19, 2012. Published January 14, 2013.} \subjclass[2000]{35B45, 35B65, 35Q30, 76D05} \keywords{Blow-up; decay; viscoelastic equation} \begin{abstract} In this article we investigate a nonlinear viscoelastic equation that admits blow-up and decay. First, we establish blow-up results for this equation, even for vanishing initial energy. Then, we show that the solutions decay under suitable conditions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \def\p{{\partial}} \def\n{\nabla} \def\L{{\Lambda}} \def\D{{\Delta}} \section{Introduction} In this article, we consider the viscoelastic equation $$\label{1.1} \begin{gathered} u_{tt}-\Delta u+\int^t_0 g(t-\tau)\Delta u(\tau)d\tau +u_t=u|u|^{p-1}, \quad (x,t)\in\Omega\times(0,\infty), \\ u(x,t)=0, \quad x\in\partial\Omega, t\geq 0,\\ u(x,0)=u_0(x), \quad u_t(x,0)=u_1(x), \quad x\in\Omega. \end{gathered}$$ where $\Omega$ is a bounded domain of $\mathbb{R}^n$ $(n\geq 1)$ with a smooth boundary $\partial\Omega$, $p>1$, and $g$ is a positive nonincreasing function. There have been extensive studies on some special cases of this equation and the physical background is also given in these works; see \cite{B,C,J,L,L1,L2,M4,V,X,B1,Z2} and references therein. For instance, the equation without $u_t$ is studied in \cite{B}, the local existence theorem is established, and for certain initial data and suitable conditions on $g$ and $p$, that this solution is global with energy which decays exponentially or polynomially depending on the rate of the decay of the relaxation function $g$. In the absence of the viscoelastic term $(g = 0)$, for instance, the equation $$u_{tt}-\Delta u+a u_t|u_t|^m=b u|u|^{\gamma}, \quad (x,t)\in\Omega\times(0,\infty),$$ we know that the source term $b u|u|^{\gamma}(\gamma>0)$ causes finite-time blow-up of solutions with negative initial energy when $a=0$, cf. \cite{J}. The interaction between the damping and the source terms was first considered by Levine \cite{L1,L2} for the linear damping case $(m=0)$. He showed that solutions with negative initial energy blow up in finite time. Recently, In \cite{Z2}, it is proved that the solution blows up in finite time even for vanishing initial energy. Another case with time dependent damping $b(t)u_t$ is studied in \cite{N3}. Georgiev and Todorova \cite{V} extended Levine¡¯s result to the nonlinear damping case $(m>0)$. In \cite{C}, it is showed that the solution blows up in finite time even for vanishing initial energy. We mention the work of Liu and Zhou \cite{L}, the equation $$u_{tt}-\Delta u=a^{-k} |u|^{\gamma}, \quad (x,t)\in \mathbb{R}^n\times(0,\infty),$$ is studied, it is proved that the solutions blow up in finite time with more relaxed initial data and extended index $\gamma$. For the problem \eqref{1.1} in $\mathbb{R}^n$, Mohammad Kafinia and Salim Messaoudib in \cite{K} give a finite-time blow-up result under suitable conditions on the initial data and the relaxation function, this work extend the result of \cite{Z1}, established for the wave equation, to the problem \eqref{1.1} in $\mathbb{R}^n$. In this paper we improve the result of blow-up in \cite{K}, and discuss the phenomenon of decay for the solution of equation \eqref{1.1}. This is an important breakthrough, since it is only well known that the solution blows up in finite time if the initial energy is negative from all the previous literature. Now, we list some notation that will be used in our paper. Use $\|\cdot\|_p$ to denote the $L^p(\mathbb{R}^n)$ norm. Throughout this paper, $C$ denotes a generic positive constant (generally large), it may be different from line to line. The remainder of the paper will be organized as follows. In the next section, we review some preliminaries that will be used in the proof of our main theorems. Then, the blow-up phenomenon will be considered in Section \ref{section3}. In the last Section, we discuss the decay of the solution to equation \eqref{1.1}. \section{Preliminaries} In this section we review some preliminaries that will be used in the proof of our main theorems. Throughout this paper, $\frac{n+2}{[n-2]_+}= \begin{cases} \infty, & n=1, 2, \\ \frac{n+2}{n-2}, & n\geq3. \end{cases}$ The relaxation function $g$ satisfies: \begin{itemize} \item[(H1)] $g:\mathbb{R}_+\to \mathbb{R}_+$ is a differentiable function such that $g(0)>0, \quad 1-\int^\infty_0 g(\tau)d\tau=l>0, \quad t\geq 0.$ \item[(H2)] There exists a positive differentiable function $\xi(t)$ such that $%\label{2.0} g'(t)\leq-\xi(t)g(t), \quad t\geq 0.$ and $\big|\frac{\xi'(t)}{\xi(t)}\big|\leq k, \xi(t)>0, \xi'(t)\leq 0, t>0.$ \end{itemize} \begin{remark} \label{rmk2.1} \rm Since $\xi$ is nonincreasing, then $\xi(t)\leq\xi(0)=M$. \end{remark} The embedding $H_0^1(\Omega)\hookrightarrow L^q(\Omega)$ for $2\leq q\leq \frac{2n}{n-2}$, if $n\geq 3$ and $q\geq 2$, if $n=1,2$; $L^r(\Omega)\hookrightarrow L^q(\Omega)$ for $q1$, such that $$\label{2.1} \|u\|_{p+1}^s\leq C\left(\|\n u\|_2^2+\|u\|^{p+1}_{p+1} \right)\quad \text{with } 2\leq s\leq p+1,$$ for any $u$ being a solution of \eqref{1.1} on $[0,T)$. Consequently, $$\label{2.5} \|u\|_{p+1}^s \leq C\left(H(t)+\|u_t\|^2_2+(g\circ\n u)(t)+\|\n u\|_2^2 \right)\quad \text{with }2\leq s\leq p+1,$$ on $[0,T)$ and here $H(t):=-E(t)$. \end{lemma} \begin{proof} If $\|u\|_{p+1} \leq 1$, the estimate $\|u\|_{p+1}^s \leq \|u\|_{p+1}^2 \leq B^2\|\n u\|_2^2$ is true. If $\|u\|_{p+1}>1$, we have $\|u\|_{p+1}^s \leq \|u\|_{p+1}^{p+1}$. Combining the two inequalities we obtain \eqref{2.1}. Note that \eqref{2.5} follows from \eqref{2.1} and the definition of energy corresponding to the solution. \end{proof} \section{Blow-up phenomenon}\label{section3} \begin{theorem} \label{theorem31} Assume that {\rm (H1), (H2)} hold, $10$ and we have $H^\alpha(t)\|u\|^2_2\leq C(\frac{1}{p+1})^\alpha \|u\|^{2+\alpha (p+1)} _{p+1}.$ Then \begin{align*} L'(t)&\geq \Big(1-\alpha-\frac{\epsilon}{2k}\Big)H^{-\alpha}(t)\|u_t\|^2_2 +\Big[p+1-\frac{kC}{2}\Big(\frac{1}{p+1}\Big)^\alpha\Big]\epsilon H(t) \\ &\quad +\Big[\frac{p-1}{2}\Big(1-\int^t_0g(\tau)d\tau\Big)-\delta -\frac{kC}{2}\Big(\frac{1}{p+1}\Big)^\alpha\Big]\epsilon\|\n u\|^2_2 \\ &\quad +\Big[\frac{p+1}{2}-\frac{1}{4\delta}\int^t_0 g(\tau)d\tau -\frac{kC}{2}\Big(\frac{1}{p+1}\Big)^\alpha\Big]\epsilon(g\circ\n u)(t) \\ &\quad +\Big[\frac{p+3}{2}-\frac{kC}{2}\Big(\frac{1}{p+1}\Big)^\alpha\Big] \epsilon\|u_t\|^2_2. \end{align*} According to the hypothesis in Theorem 3.1 and take $k$ and $\delta$ to be small enough such that \begin{gather*} \frac{p-1}{2}\Big(1-\int^t_0g(\tau)d\tau\Big)-\delta -\frac{kC}{2}\Big(\frac{1}{p+1}\Big)^\alpha> 0,\\ \frac{p+1}{2}-\frac{1}{4\delta}\int^t_0 g(\tau)d\tau -\frac{kC}{2}\Big(\frac{1}{p+1}\Big)^\alpha>0. \end{gather*} Choose $\epsilon$ ($k$ is fixed) small enough such that $1-\alpha-\frac{\epsilon}{2k}\geq 0, \quad L(0)=H^{1-\alpha}(0)+\epsilon \int_\Omega u_0 u_1dx > 0.$ Then, we can deduce that $L'(t) \geq C[H(t)+\|u_t\|^2_2+\|\n u\|^2_2+(g \circ\n u)(t)].$ Thanks to H\"{o}lder and Young inequality, we obtain $$\label{3.33} \begin{split} \Big|\int_\Omega uu_tdx\Big|^{1/(1-\alpha)} &\leq \|u\|^{1/(1-\alpha)}_2 \|u_t\|^{1/(1-\alpha)}_2 \leq C\|u\|^{1/(1-\alpha)}_{p+1} \|u_t\|^{1/(1-\alpha)}_2 \\ &\leq C(\|u\|^s_{p+1}+ \|u_t\|^2_2) \\ &\leq C\left(H(t)+\|u_t\|^2_2+(g\circ\n u)(t)+\|\n u\|_2^2\right), \end{split}$$ where $2\leq s= \frac{2}{1-2\alpha}\leq p+1$. Hence, \begin{align*} L^{1/(1-\alpha)}(t) &= \Big(H^{1- \alpha }(t)+ \epsilon\int _\Omega uu_t dx\Big)^{1/(1-\alpha)}\\ &\leq 2^{1/(1-\alpha)}\Big(H(t) +\Big|\int _\Omega uu_t dx\Big|^{1/(1-\alpha)} \Big) \\ &\leq C\left(H(t)+\|u_t\|^2_2+(g\circ\n u)(t)+\|\n u\|_2^2\right), \end{align*} which implies that $L'(t)\geq \lambda L^{1/(1-\alpha)}(t)$, where $\lambda$ is a constant depending on $C$, $p$, $\alpha$ and $\epsilon$. Therefore $L(t)\geq(L^{\frac{-\alpha}{1-\alpha}}(0) + \frac{-\alpha}{1-\alpha}\lambda t)^{-\frac{1-\alpha}{\alpha}}.$ So $L(t)$ approaches infinite as $t$ tends to $(1-\alpha)/\big(\alpha \lambda L^{\frac{\alpha}{1-\alpha}}(0)\big)$. This completes the proof. \end{proof} To obtain another blow-up result we first give the following lemma. \begin{lemma} \label{lem3.2} Assume that {\rm (H1), (H2)} hold, additionally, assume that $\|u_0\|_{p+1}>\lambda_0\equiv B_0^{\frac{-2}{p-1}}, \quad E(0)\lambda_0, \quad \|\n u\|_2 >B_0^{\frac{-(p+1)}{p-1}},\quad\text{for all }t \geq 0,$ where $B_0=\frac{B}{l^{1/2}}$ for $\|u\|_{p+1}\leq\ B\|\n u\|_2$. \end{lemma} \begin{proof} From \eqref{2.10} and the hypothesis, we know that \begin{align*} E(t)&=\frac{1}{2} \|u_t\|^2_2+ \frac{1}{2} \Big(1-\int^t_0 g(\tau)d\tau\Big)\|\n u\|^2_2 +\frac{1}{2}(g\circ\n u)(t)-\frac{1}{p+1}\|u\|^{p+1}_{p+1} \\ &\geq \frac{1}{2}\Big(1-\int^t_0 g(\tau)d\tau\Big)\|\n u\|^2_2 -\frac{1}{p+1}\|u\|^{p+1}_{p+1} \\ &\geq \frac{l}{2}\|\n u\|^2_2-\frac{1}{p+1}\|u\|^{p+1}_{p+1} \geq\frac{1}{2B^2_0}\|u\|^2_{p+1}-\frac{1}{p+1}\|u\|^{p+1}_{p+1}. \end{align*} Set $h(\xi)= \frac{1}{2B_0^2}\xi^2- \frac{1}{p+1}\xi^{p+1}$, $\xi \geq 0$. Then $h(\xi)$ satisfies \begin{itemize} \item $h(\xi)$ is strictly increasing on $[0,\lambda_0)$; \item $h(\xi)$ takes its maximum value $(\frac{1}{2}-\frac{1}{p+1})B_0^{\frac{-2(p+1)}{p-1}}$ at $\lambda_0$; \item $h(\xi)$ is strictly decreasing on $(\lambda_0, \infty)$. \end{itemize} Since $E_0 > E(0)\geq E(t) \geq h(\|u\|_{p+1})$ for all $t\geq0$, there is no time $t^*$ such that $\|u(\cdot,t^*)\|_{p+1}=\lambda_0$. By the continuity of the $\|u(\cdot,t)\|_{p+1}-norm$ with respect to the time variable, one has $\|u(\cdot,t)\|_{p+1} > \lambda_0=B_0^{\frac{-2}{p-1}}\quad \text{for all t \geq 0},$ and consequently, $\|\n u(\cdot,t)\|_2 \geq \frac{1}{l^{1/2}B_0}\|u(\cdot,t)\|_{p+1} >\frac{1}{l^{1/2}}B_0^{\frac{-(p+1)}{p-1}}>B_0^{\frac{-(p+1)}{p-1}}.$ This completes the proof. \end{proof} \begin{theorem} \label{thm3.3} Suppose that{\rm (H1), (H2)} hold, $1\lambda _0$ and $E(0)\leq E_0$. Then the solution of \eqref{1.1} blows up in finite time. \end{theorem} \begin{proof} Set $G(t)=E_0+H(t)$, then $G'(t)=-\frac{1}{2}(g'\circ\n u)(t)+\frac{1}{2}g(t)\|\n u\|^2_2+\|u_t\|^2_2\geq 0,$ from which we obtain \begin{align*} 0< G(t) &= E_0+H(t)=\big(\frac{1}{2}-\frac{1}{p+1}\big)B_0^ {\frac{-2(p+1)}{p-1}}+H(t) \\ &< \big(\frac{1}{2}-\frac{1}{p+1}\big)\|\n u\|^2_2+H(t) 0, then $I(u(t))>0$, for all $t>0$. Here $C_e$ is given in \eqref{em}. \end{lemma} \begin{proof} Since $I(u_0)>0$, there exists T_m0, \quad \forall t\in[0,T_m], \] which gives \begin{align*} &\frac{1}{2}\Big(1-\int^t_0 g(\tau)d\tau\Big)\|\n u\|^2_2 +\frac{1}{2}(g\circ\n u)(t)-\frac{1}{p+1}\|u\|^{p+1}_{p+1} \\ &= \frac{p-1}{2(p+1)}\Big[\Big(1-\int^t_0 g(\tau)d\tau\Big)\|\n u\|^2_2 +(g\circ\n u)(t)\Big]+\frac{1}{p+1}I(t) \\ &\geq \frac{p-1}{2(p+1)}\Big[\Big(1-\int^t_0 g(\tau)d\tau\Big)\|\n u\|^2_2 +(g\circ\n u)(t)\Big]. \end{align*} So we have $$\label{2.7} l\|\n u\|^2_2\leq\Big(1-\int^t_0 g(\tau)d\tau\Big)\|\n u\|^2_2 \leq\frac{2(p+1)}{p-1}E(t)\leq\frac{2(p+1)}{p-1}E(0).$$ By using (H1), \eqref{2.3} and \eqref{2.7}, we obtain $\|u\|^{p+1}_{p+1}\leq C_e^{p+1}\|\n u\|^{p+1}_2\leq \beta l\|\n u\|^2_2 <\left(1-\int^t_0 g(\tau)d\tau\right)\|\n u\|^2_2$ Hence, $I(t)=\left(1-\int^t_0 g(\tau)d\tau\right)\|\n u\|^2_2 +(g\circ\n u)(t)-\|u\|^{p+1}_{p+1}>0,\quad \forall t\in [0,T_m].$ By repeating this process, and using that $\lim_{t\to T_m}\frac{C_e^{p+1}}{l}\Big(\frac{2(p+1)E(u,u_t)}{(p-1)l}\Big)^{(p-1)/2}\leq\beta<1,$ we show thatT_m$is extended to$T$. \end{proof} To establish the decay rate, we use the functional $$\label{3.1} F(t)=E(t)+\epsilon_1\Psi(t)+\epsilon_2\Phi(t),$$ where$\epsilon_1$and$\epsilon_2$are positive constants and $\Psi(t)=\xi(t)\int_\Omega uu_tdx, \quad \Phi(t)=-\xi(t)\int_\Omega u_t\int_0^tg(t-\tau)(u(t)-u(\tau))\,d\tau\,dx.$ This functional, for$\xi(t)=1$, was first introduced in \cite{B} and \cite{B1}. Now, let us consider some useful properties of this functional. \begin{lemma} \label{lem4.2} Assume that$u(x, t)$is the solution of \eqref{1.1} and that \eqref{2.3} holds. Then there exists$k_1<1$and$k_2>1such that $$\label{4.2} k_1E(t)\leq F(t)\leq k_2E(t).$$ \end{lemma} \begin{proof} Using Young, Schwarz and Poincar\'e inequality, we obtain \begin{gather} \label{3.4} \int_\Omega uu_tdx\leq\frac{C_*^2}{2}\|\n u\|^2_2+\frac{1}{2}\|u_t\|^2_2,\\ \label{3.5} \int_\Omega u_t\int_0^tg(t-\tau)(u(t)-u(\tau))\,d\tau\,dx \leq \frac{1}{2}\|u_t\|^2_2+\frac{1}{2}(1-l)C_*^2(g\circ\n u)(t). \end{gather} Using \eqref{3.4} and \eqref{3.5}, we have \begin{align*} k_2E(t)-F(t) &\geq \Big[\Big(\frac{k_2-1}{2}-\frac{k_2-1}{p+1} \Big)l-\frac{\epsilon_1C_*^2M}{2}\Big]\|\n u\|^2_2 \\ &\quad +\frac{1}{2}\{k_2-1-(\epsilon_1+\epsilon_2)M\}\|u_t\|^2_2 +\frac{k_2-1}{p+1}I(t) \\ &\quad+\Big[\frac{k_2-1}{2}-\frac{k_2-1}{p+1}-\frac{\epsilon_2(1-l) C_*^2M}{2}\Big](g\circ\n u)(t). \end{align*} Similarly, \begin{align*} F(t)-k_1E(t) &\geq \Big[\Big(\frac{1-k_1}{2}-\frac{1-k_1}{p+1} \Big)l-\frac{\epsilon_1C_*^2M}{2}\Big]\|\n u\|^2_2 \\ &\quad +\frac{1}{2}[1-k_1-(\epsilon_1+\epsilon_2)M]\|u_t\|^2_2+\frac{1-k_1}{p+1} I(t) \\ &\quad+\Big[\frac{1-k_1}{2}-\frac{1-k_1}{p+1}-\frac{\epsilon_2(1-l) C_*^2M}{2}\Big](g\circ\n u)(t). \end{align*} By choosing\epsilon_1$and$\epsilon_2$small enough, such that$k_2E(t)-F(t)\geq 0$and$F(t)-k_1E(t)\geq 0$, we complete the proof. \end{proof} \begin{lemma} \label{lem4.3} Let {\rm (H1) and (H2)} hold, and$p\leq\frac{n+2}{[n-2]_+}$. Assume that$(u_0,u_1)\in H_0^1(\Omega)\times L^2(\Omega)$and$uis the solution of \eqref{1.1}. Then $$\label{3.8} \begin{split} \Psi'(t) &\leq \Big(1+\frac{(1-k)(1+k)C_*^2}{l}\Big)\xi(t)\|u_t\|^2_2 +\frac{1-l}{2l}\xi(t)(g\circ\n u)(t) \\ &\quad-\frac{l}{4}\xi(t)\|\n u\|^2_2+\xi(t)\|u\|^{p+1}_{p+1} \end{split}$$ \end{lemma} \begin{proof} By a direct computation, we have $$\label{3.9} \begin{split} \Psi'(t) &= \xi(t)\Big(\|u_t\|^2_2+\|u\|^{p+1}_{p+1}-\|\n u\|^2_2 +\int_\Omega\n u(t)\int_0^tg(t-\tau)\n u(\tau)\,d\tau\,dx \\ &\quad -\int_\Omega uu_tdx\Big)+\xi'(t)\int_\Omega uu_tdx \\ &:=\xi(t)\left(\|u_t\|^2_2+\|u\|^{p+1}_{p+1}-\|\n u\|^2_2+A_1-A_2 \right)+\xi'(t)A_2. \end{split}$$ By Young, Schwarz and Poincar\'{e} inequality, we have \begin{gather} A_1\leq \frac{1}{2}\|\n u\|^2_2+\frac{1}{2}\big(1+\frac{1}{\eta}\big) (1-l)(g\circ\n u)(t) +\frac{1}{2}(1+\eta)(1-l)^2\|\n u\|^2_2, \label{3.10} \\ A_2 \leq \alpha C_*^2\|\n u\|^2_2+\frac{1}{4\alpha}\|u_t\|^2_2.\label{3.11} \end{gather} Combining \eqref{3.9} and \eqref{3.10} with \eqref{3.11} yields \begin{align*} \Psi'(t) &\leq \big(1+\frac{1-k}{4\alpha}\big)\xi(t)\|u_t\|^2_2 +\frac{1}{2}\big(1+\frac{1}{\eta}\big)(1-l)\xi(t)(g\circ\n u)(t) \\ &\quad -\Big[\frac{1}{2}-\frac{(1+\eta)(1-l)^2}{2}-(1+k)\alpha C_*^2\Big] \xi(t)\|\n u\|^2_2+\xi(t)\|u\|^{p+1}_{p+1}. \end{align*} We choose\eta=l/(1-l)$and$\alpha=l/(4(1+k)C_*^2)$; then \eqref{3.8} is true. \end{proof} \begin{lemma} \label{lem4.4} Let {\rm (H1)} and {\rm (H2)} hold,$p\leq\frac{n+2}{[n-2]_+}$,$(u_0,u_1)\in H_0^1(\Omega)\times L^2(\Omega)$and$u$is the solution of \eqref{1.1}. Then $$\label{3.14} \begin{split} \Phi'(t) &\leq \delta\Big[1+2(1-l)^2+C_e^{2p} \Big(\frac{2(p+1)E(0)}{l(p-1)}\Big)^{p-1}\Big]\xi(t)\|\n u\|^2_2 \\ &\quad-\frac{g(0)C_*^2}{4\delta}\xi(t)(g'\circ\n u)(t) +\Big[\big(2\delta +\frac{1}{2\delta}\big)(1-l) +\frac{(2+k)(1-l)C_*^2}{4\delta}\Big] \\ &\quad \times\xi(t)(g\circ\n u)(t)+\Big[\delta(2+k)-\int_0^tg(\tau)d\tau \Big]\xi(t)\|u_t\|^2_2. \end{split}$$ \end{lemma} \begin{proof} Straightforward computations show that $$\label{3.15} \begin{split} \Phi'(t)&= \xi(t)\int_\Omega\n u\int_0^tg(t-\tau)(\n u(t)-\n u(\tau)) \,d\tau\,dx \\ &\quad -\xi(t)\int_\Omega\Big(\int_0^tg(t-\tau)\n u(\tau)d\tau\Big) \Big(\int_0^tg(t-\tau)(\n u(t)-\n u(\tau))d\tau\Big)dx \\ &\quad +\xi(t)\int_\Omega u_t\int_0^tg(t-\tau)(u(t)-u(\tau))\,d\tau\,dx \\ &\quad -\xi(t)\int_\Omega u|u|^{p-1}\int_0^tg(t-\tau)(u(t)-u(\tau))\,d\tau\,dx \\ &\quad -\xi(t)\int_\Omega u_t\int_0^tg'(t-\tau)(u(t)-u(\tau))\,d\tau\,dx -\xi(t)\Big(\int_0^tg(\tau)d\tau\Big)\|u_t\|^2_2 \\ &\quad -\xi'(t)\int_\Omega u_t\int_0^tg(t-\tau)(u(t)-u(\tau))\,d\tau\,dx \\ &:=\xi(t)\Big[I_1+I_2+I_3+I_4+I_5-\Big(\int_0^tg(\tau)d\tau\Big)\|u_t\|^2_2\Big] -\xi'(t)I_3. \end{split}$$ By Young and Poincar\'{e} inequality, we have \begin{gather} I_1\leq \delta\|\n u\|^2_2+\frac{1-l}{4\delta}(g\circ\n u)(t),\label{3.16}\\ I_2\leq \Big(2\delta+\frac{1}{4\delta}\Big)(1-l)(g\circ\n u)(t) +2\delta(1-l)^2\|\n u\|^2_2,\label{3.17}\\ I_3\leq \delta\|u_t\|^2_2+\frac{C_*^2(1-l)}{4\delta}(g\circ\n u)(t),\label{3.18}\\ I_4\leq \delta C_e^{2p}\Big(\frac{2(p+1)E(0)}{l(p-1)}\Big)^{p-1}\|\n u\|^2_2 +\frac{(1-l)C_*^2}{4\delta}(g\circ\n u)(t),\label{3.19}\\ I_5\leq \delta\|u_t\|^2_2-\frac{g(0)C_*^2}{4\delta}(g'\circ\n u)(t).\label{3.20} \end{gather} Combining \eqref{3.15}-\eqref{3.20}, we have the required estimate \eqref{3.14}. \end{proof} We are ready to give our decay result. \begin{theorem} \label{thm4.5} Suppose that {\rm (H1), (H2)} and \eqref{2.3} hold,$p\leq\frac{n+2}{[n-2]_+}$,$(u_0,u_1)\in H_0^1(\Omega)\times L^2(\Omega)$. Then there exists positive constants$\alpha$and$\lambda$such that the solution of \eqref{1.1} satisfies $E(t)\leq \alpha e^{-\lambda\int_{t_0}^t \xi(\tau)d\tau},\quad t\geq t_0.$ \end{theorem} \begin{proof} Since$g$is positive, continuous and$g(0)>0$, then for any$t_0>0$, we have $$\label{3.21} \int_0^tg(\tau)d\tau\geq\int_0^{t_0}g(\tau)d\tau=g_0>0, \quad\forall t\geq t_0.$$ Combining \eqref{2.2}, \eqref{3.1}, \eqref{3.8}, \eqref{3.14} and \eqref{3.21}, for$t\geq t_0$, we have $$\label{4.17} \begin{split} F'(t) &\leq -\Big\{\epsilon_2[g_0-\delta(2+k)]-\epsilon_1 \Big(1+\frac{(1-k)(1+k)C_*^2}{l}\Big)\Big\}\xi(t)\|u_t\|^2_2 \\ &\quad -\Big\{\frac{\epsilon_1l}{4}-\epsilon_2\delta \Big[1+2(1-l)^2+C_e^{2p} \Big(\frac{2(p+1)E(0)}{l(p-1)}\Big)^{p-1}\Big]\Big\} \xi(t)\|\n u\|^2_2 \\ &\quad +\Big\{\frac{\epsilon_1(1-l)}{2l}+\epsilon_2 \Big[\Big(2\delta+\frac{1}{2\delta}\Big)(1-l) +\frac{(2+k)(1-l)C_*^2}{4\delta}\Big]\Big\}\xi(t)(g\circ\n u)(t) \\ &\quad +\Big(\frac{1}{2}-\frac{\epsilon_2g(0)C_*^2M}{4\delta}\Big)(g'\circ\n u)(t) +\epsilon_1\xi(t)\|u\|^{p+1}_{p+1} \\ &:=-J_1\xi(t)\|u_t\|_2^2-J_2\xi(t)\|\n u\|_2^2+J_3\xi(t)(g\circ\n u)(t) \\ &\quad +J_4(g'\circ\n u)(t)+\epsilon_1\xi(t)\|u\|^{p+1}_{p+1}. \end{split}$$ We choose suitable constants$\epsilon_1$and$\epsilon_2$satisfying $\frac{\epsilon_1\big(1+\frac{(1-k)(1+k)C_*^2}{l}\big)}{g_0-\delta(2+k)} <\epsilon_2<\frac{\epsilon_1l}{4\delta\big[1+2(1-l)^2+C_e^{2p} \big(\frac{2(p+1)E(0)}{l(p-1)}\big)^{p-1}\big]}$ and$\delta$,$\epsilon_1$small enough, such that $g_0-(2+k)\delta>\frac{1}{2}g_0,\quad J_1>0,\quad J_2>0,\quad k_3:=J_4-J_3>0,$ which imply $J_4(g'\circ\n u)(t)+J_3\xi(t)(g\circ\n u)(t)\leq-k_3\xi(t)(g\circ\n u)(t).$ Applying \eqref{4.2} and \eqref{4.17} yields $F'(t)\leq-\gamma\xi(t)E(t)\leq\frac{-\gamma}{k_2}\xi(t)F(t).$ Therefore, after integrating the above inequality and using \eqref{4.2} again, we obtain the desire result. \end{proof} \subsection*{Acknowledgments} This work is partially supported by grant T200905 from the Zhejiang Innovation Project, and grant 11226176 from the NSFC. The authors want thank the anonymous referee for the careful reading of the original manuscript and the helpful suggestions. \begin{thebibliography}{00} \bibitem{J} J. Ball; \emph{Remarks on blow up and nonexistence theorems for nonlinear evolutions equations}, Q. J. Math. Oxford 28 (1977) 473--486. \bibitem{B1} S. Berrimi, S. A. Messaoudi; \emph{Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping}. Electron, J. Differential Equations 2004 (2004) no. 88, 1-10. \bibitem{B} S. Berrimi, S.A. Messaoudi; \emph{Existence and decay of solutions of a viscoelastic equation with a nonlinear source}, Nonlinear Anal. 64 (2006) 2314--2331. \bibitem{C} W. Chen, Y. Zhou; \emph{Global nonexistence for a semilinear Petrovsky equation}, Nonlinear Anal. 70 (2009) 3203--3208. \bibitem{V} V. Georgiev, G. Todorova; \emph{Existence of solutions of the wave equation with nonlinear damping and source terms}, J. Differential Equations 109 (1994) 295-308. \bibitem{K} M. Kafini, S. A. Messaoudi; \emph{A blow-up result in a Cauchy viscoelastic problem}, Appl. Math. Lett. 21 (2008) 549--553. \bibitem{L1} H. A. Levine; \emph{Instability and nonexistence of global solutions of nonlinear wave equation of the form$Pu_{tt} = Au + F(u)$}, Trans. Amer. Math. Soc. 192 (1974) 1-21. \bibitem{L2} H. A. Levine; \emph{Some additional remarks on the nonexistence of global solutions to nonlinear wave equation}, SIAM J. Math. Anal. 5 (1974) 138-146. \bibitem{L} X. Liu, Y. Zhou; \emph{Global nonexistence of solutions to a semilinear wave equation in the Minkowski space}, Appl. Math. Lett. 21 (2008) 849--854. \bibitem{M4} S.A. Messaoudi; \emph{General decay of the solution energy in a viscoelastic equation with a nonlinear source}, Nonlinear Anal. 69 (2008) 2589--2598. \bibitem{N3} K. Nishihara, J. Zhai; \emph{Asymptotic behaviors of solutions for time dependent damped wave equations}, J. Math. Anal. Appl. 360 (2009) 412--421. \bibitem{X} Y. Xiong; \emph{Blow-up and polynomial decay of solutions for a viscoelastic equation with a nonlinear source}, Z. Anal. Anwend. 31 (2012), no. 3, 251--266. \bibitem{Z1} Y. Zhou; \emph{Global existence and nonexistence for a nonlinear wave equation with damping and source terms}, Math. Nachr. 278 (2005) 1341--1358. \bibitem{Z2} Y. Zhou; \emph{A blow-up result for a nonlinear wave equation with damping and vanishing initial energy in$\mathbb{R}^N\$}, Appl. Math. Lett. 18 (2005) 281--286. \end{thebibliography} \end{document}