\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 105, 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 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/105\hfil Blow-up of solutions] {Blow-up of solutions for a system of nonlinear wave equations with nonlinear damping} \author[S.-T. Wu \hfil EJDE-2009/105\hfilneg]{Shun-Tang Wu} \address{Shun-Tang Wu \\ General Education Center\\ National Taipei University of Technology \\ Taipei, 106, Taiwan} \email{stwu@ntut.edu.tw} \thanks{Submitted February 17, 2009. Published September 1, 2009.} \subjclass[2000]{35L70} \keywords{Nonlinear damping; wave equation; blow-up; lifespan} \begin{abstract} We study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain $\Omega$ with the initial and Dirichlet boundary conditions. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In this article we shall consider the following initial-boundary value problem for a system of nonlinear wave equations: \begin{gather} \square u+|u_{t}| ^{p-1}u_{t}+m_{1}^{2}u =4\lambda(u+\alpha v)^{3}+2\beta uv^{2}\quad\text{in }\Omega\times[0,T),\label{e1.1}\\ \square v+|v_{t}| ^{q-1}v_{t}+m_{2}^{2}v =4\alpha\lambda(u+\alpha v)^{3}+2\beta vu^{2}\quad\text{in }\Omega\times[0,T),\label{e1.2} \end{gather} with initial conditions \begin{gather} u(x,0) =u_{0}(x),\quad u_{t}( x,0)=u_{1}(x),\quad x\in\Omega,\label{e1.3}\\ v(x,0) =v_{0}(x),\quad v_{t}(x,0)=v_{1}(x),\quad x\in\Omega,\label{e1.4} \end{gather} and boundary conditions \begin{gather} u(x,t)=0,\quad x\in\partial\Omega,\; t>0,\label{e1.5}\\ v(x,t) =0,\quad x\in\partial\Omega,\; t>0,\label{e1.6} \end{gather} where $\square=\frac{\partial^{2}}{\partial t^{2}}-\Delta$, $\Delta=\sum _{j=1}^{3}\frac{\partial^{2}}{\partial x_{j}^{2}}$ and $\Omega\subset \mathbb{R}^{3}$ is a bounded domain with a smooth boundary $\partial\Omega$ so that Divergence theorem can be applied and $\lambda,\beta$ and $\alpha$ are real numbers, and $p$, $q\geq1$, $T>0$. The initial-boundary value problem for a single wave equation: $$u_{tt}-\Delta u(t)+a|u_{t}(t)|^{p-1}u_{t}(t)=f(u),\label{e1.7}$$ where $a>0$, $p\geq1$, was considered by many authors. For $f(u)=|u|^{m-1}u$, $m>1$, this model was first studied by Levine \cite{l1,l2} in the linear case $(p=1)$. He showed that solutions with negative initial energy blow up in finite time. When $p=1$, Ikehata \cite{i2} proved that for sufficiently small initial data, the trajectory $(u(t),v(t))$ goes to $(0,0)$ in $H_{0}^{1}(\Omega)\times L^{2}(\Omega)$ as $t\rightarrow\infty$. Georgiev and Todorova \cite{g1} extended Levine's result to nonlinear case $(p>1)$. They showed that solutions continue to exist globally if $p\geq m$ and blow up in finite time if $p1$. In unbounded domain, for $f(u)=-\lambda (x)^{2}u+|u|^{m-1}u$, $m>1$, here $\lambda(x)$ satisfies some decay conditions, there are some results about global existence and asymptotic behavior in \cite{n1}. Aassila \cite{a1} treated \eqref{e1.7} for $f(u)=-u+|u|^{m-1}u$, $m>1$, and gave the global existence and energy decay property. Reed \cite{r1} proposed this interesting problem of \eqref{e1.1}-\eqref{e1.6} without damping terms in \eqref{e1.1} and \eqref{e1.2}. As a model it describes the interaction of scalar fields $u,v$ of mass $m_{1},m_{2}$ respectively. This system defines the motion of charged mesons in an electromagnetic field which was first introduced by Segal \cite{s1}. Later, J\"{o}rgens \cite{j1}, Makhankov \cite{m1}, and Medeiros and Menzala \cite{m2} studied such systems to find the existence of weak solutions of the mixed problem in a bounded domain. Further generalizations are also given in [12,13] by Galerkin method. Recently, the existence of global and nonglobal solutions of a system of semilinear wave equations without dissipative terms were discussed in \cite{l3,l4}. In this paper we are interested in the blow-up behavior of solutions for a system \eqref{e1.1}-\eqref{e1.6} in a bounded domain $\Omega$ in $\mathbb{R}^{3}$. This work improves an earlier work \cite{l4}, in which similar results have been established for \eqref{e1.1}-\eqref{e1.6} in the absence of the damping terms. The paper is organized as follows. In section 2, we give some lemmas which will be used later, and we mention the local existence Theorem \ref{thm2.4}. In section 3, we first define an energy function $E(t)$ by \eqref{e3.1} and show that it is a nonincreasing function of $t$. Then, we discuss the blow-up properties of \eqref{e1.1}-\eqref{e1.6} in two cases. In first case, $p=q=1$, the main result is given in Theorem \ref{thm3.4}, which contains the estimates of upper bound of the blow-up time. In second case, $10$ and $B(t)\in C^{2}(0,\infty)$ be a nonnegative function satisfying $$B''(t)-4(\delta+1)B'(t)+4(\delta+1)B(t)\geq0.\label{e2.1}$$ \textit{If } $$B'(0)>r_{2}B(0)+K_{0},\label{e2.2}$$ with $r_{2}=2(\delta+1)-2\sqrt{(\delta+1)\delta}$, then $B'(t)>K_{0}$ for $t>0$, where $K_{0}$ is a constant. \end{lemma} \begin{lemma}[\cite{l3}] \label{lem2.3} If $J(t)$ is a nonincreasing function on [$t_{0}, \infty)$ and satisfies the differential inequality $$J'(t)^{2}\geq a+bJ(t)^{2+\frac{1}{\delta}},\quad \textit{for }t\geq t_{0},\label{e2.3}$$ where $a>0,b\in \mathbb{R}$, then there exists a finite time $T^{\ast}$ such that $\lim_{t\to T^{\ast-}}J(t)=0\,.$ Upper bounds for $T^{\ast}$ are estimated as follows: \begin{itemize} \item[(i)] If $b<0$, then $T^{\ast}\leq t_{0}+\frac{1}{\sqrt{-b}}\ln\frac{\sqrt{-a/b}} {\sqrt{-a/b}-J(t_{0})}.$ \item[(ii)] If $b=0$, then $T^{\ast}\leq t_{0}+\frac{J(t_{0})}{J'(t_{0})}.$ \item[(iii)] If $b>0$, then $T^{\ast}\leq\frac{J(t_{0})}{\sqrt{a}} \quad\text{or}\quad T^{\ast}\leq t_{0}+2^{(3\delta+1)/(2\delta)}\frac{\delta c}{\sqrt{a} }\{1-[1+cJ(t_{0})]^{-1/(2\delta)}\},$ where $c=(\frac{a}{b})^{2+\frac{1}{\delta}}$. \end{itemize} \end{lemma} Now, we state the local existence result which is proved in \cite{w3}. \begin{theorem}[Local solution] \label{thm2.4} Let $p$, $q\geq 1$, and $u_{0},v_{0}\in H_{0}^{1}(\Omega)$, $u_{1},v_{1}\in L^{2}(\Omega)$, then there exists a unique local solution $(u,v)$ of \eqref{e1.1}-\eqref{e1.6} satisfying $(u,v)\in Y_{T}$, where \begin{align*} Y_{T}=\big\{&w=(u,v): w\in C([0,T];H_{0}^{1}(\Omega)\times H_{0}^{1} (\Omega)),\; w_{t}\in C([0,T];L^{2}(\Omega)\times L^{2}(\Omega)),\\ &u_{t}\in L^{p+1}(\Omega\times(0,T)),\; v_{t}\in L^{q+1}(\Omega\times(0,T)) \big\}. \end{align*} \end{theorem} \section{Blow-up property} In this section, we will discuss the blow up phenomena of two problems, where $p=q=1$ in subsection 3.1 and $1\| u_{0}\|_{2}^{2}+\|v_{0}\|_{2}^{2}$ for $t>t^{\ast}$, where $$t^{\ast}=\max\big\{ \frac{a'(0)-(\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2} ^{2})}{4(1+2\delta)E(0)},\, 0\big\} .\label{e3.9}$$ \noindent(2) If $E(0)=0$, then $a''(t)\geq0$ for $t\geq0$. If $a'(0) >\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2}^{2}$, then we have $a'(t)>\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2}^{2}$, $t\geq0$. \noindent(3) For the case that $E(0)>0$, we first note that $$2\int_{0}^{t}\int_{\Omega}uu_{t}\,dx\,dt =\| u\|_{2}^{2}-\| u_{0}\|_{2} ^{2}.\label{e3.10}$$ By H\"{o}lder inequality and Young's inequality, we have from \eqref{e3.10}, $$\| u\|_{2}^{2}\leq\| u_{0}\|_{2}^{2}+\int_{0}^{t}\| u\| _{2}^{2}dt+\int _{0}^{t}\| u_{t}\|_{2}^{2}dt.\label{e3.11}$$ Similarly, $$\| v\|_{2}^{2}\leq\| v_{0}\|_{2}^{2}+\int_{0}^{t}\| v\| _{2}^{2}dt+\int _{0}^{t}\| v_{t}\|_{2}^{2}dt.\label{e3.12}$$ By H\"{o}lder inequality, Young$'$s inequality and then using \eqref{e3.11} and \eqref{e3.12}, we have from \eqref{e3.7}, $$a'(t)\leq a(t)+\| u_{0}\| _{2}^{2}+\| v_{0}\|_{2}^{2}+\int_{\Omega }(u_{t}^{2}+v_{t} ^{2})\,dx+\int_{0}^{t}(\| u_{t}\|_{2}^{2}+\| v_{t} \|_{2}^{2})dt.\label{e3.13}$$ Hence by \eqref{e3.6} and \eqref{e3.12}, we obtain $a''(t)-4(\delta+1)a'(t)+4(\delta+1)a(t)+K_{1}\geq0,$ where $K_{1}=(4+8\delta)E(0)+4(\delta+1)(\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2}^{2})\, .$ Let $b(t)=a(t)+\frac{K_{1}}{4(1+\delta)},\quad t>0.$ Then $b(t)$ satisfies \eqref{e2.1}. By Lemma \ref{lem2.3} we see that if $$a'(0)>r_{2}\big[ a(0)+\frac{K_{1} }{4(1+\delta)}\big] +(\| u_{0} \|_{2}^{2}+\| v_{0}\| _{2}^{2}),\label{e3.14}$$ then $a'(t)>(\| u_{0}\|_{2}^{2}+\|v_{0}\|_{2}^{2})$, $t>0$, where $r_{2}$ is given in Lemma \ref{lem2.2}. Consequently, we have the following result. \begin{lemma} \label{lem3.3} Assume {\rm (A1)} and that either one of the following statements is satisfied: \begin{itemize} \item[(i)] $E(0)<0$, \item[(ii)] $E(0)=0$ and $a'(0)>\|u_{0}\|_{2}^{2} +\| v_{0}\|_{2}^{2}$, \item[(iii)] $E(0)>0$ and \eqref{e3.14} holds\,. \end{itemize} Then, $a'(t)>\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2}^{2}$ for $t>t_{0}$, where $t_{0}=t^{\ast}$ is given by \eqref{e3.9} in case (i) and $t_{0}=0$ in cases (ii) and (iii). \end{lemma} Now, we find an estimate for the life span of $a(t)$. Let $$J(t)=\big[ a(t)+(T_{1}-t)(\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2}^{2} )\big] ^{-\delta},\quad\text{for } t\in[0,T_{1}],\label{e3.15}$$ where $T_{1}>0$ is a certain constant which will be specified later. Then we have $$\begin{gathered} J'(t)=-\delta J(t)^{1+\frac{1}{\delta} } (a'(t)-\| u_{0}\|_{2}^{2}-\| v_{0}\|_{2}^{2}), \\ J''(t) =-\delta J(t)^{1+\frac{2}{\delta}}V(t),\label{e3.16} \end{gathered}$$ where \begin{aligned} V(t) &=a''(t)\left[ a( t)+(T_{1}-t)(\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2} ^{2})\right]\\ &\quad -(1+\delta)(a'(t)-\| u_{0}\|_{2}^{2}-\| v_{0}\|_{2}^{2})^{2}.\label{e3.17} \end{aligned} For simplicity of calculation, we denote \begin{gather*} P_{u} =\int_{\Omega}u^{2}\,dx,\quad P_{v}=\int_{\Omega}v^{2}\,dx,\\ Q_{u} =\int_{0}^{t}\|u\|_{2}^{2}dt,\quad Q_{v} =\int_{0}^{t}\|v\|_{2}^{2}dt,\\ R_{u} =\int_{\Omega}u_{t}^{2}\,dx,\quad R_{v}=\int_{\Omega}v_{t}^{2}\,dx,\\ S_{u} =\int_{0}^{t}\|u_{t}\|_{2}^{2}dt,\quad S_{v}=\int_{0}^{t}\|v_{t} \|_{2}^{2}dt. \end{gather*} From \eqref{e3.7}, \eqref{e3.10}, and H\"{o}lder inequality, we get \begin{aligned} a'(t)& =2\int_{\Omega}(uu_{t}+vv_{t}) \,dx +\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2}^{2}+2\int_{0}^{t} \int _{\Omega}(uu_{t}+vv_{t})\,dx\,dt \\ & \leq 2(\sqrt{R_{u}P_{u}} +\sqrt{Q_{u}S_{u}}+\sqrt{R_{v}P_{v}}+\sqrt{Q_{v} S_{v}})+\| u_{0}\|_{2}^{2} +\| v_{0}\|_{2}^{2}. \end{aligned}\label{e3.18} By \eqref{e3.6}, we have $$a''(t)\geq(-4-8\delta)E(0)+4(1+\delta)(R_{u}+S_{u}+R_{v}+S_{v}) .\label{e3.19}$$ Thus, from \eqref{e3.18}, \eqref{e3.19}, \eqref{e3.17} and \eqref{e3.15}, we obtain \begin{align*} V(t) & \geq\left[ (-4-8\delta)E( 0)+4(1+\delta)(R_{u}+S_{u}+R_{v}+S_{v}) \right] J(t)^{-1/\delta}\\ & \quad-4(1+\delta)(\sqrt{R_{u}P_{u}}+\sqrt{Q_{u}S_{u}} +\sqrt{R_{v}P_{v} }+\sqrt{Q_{v}S_{v}})^{2}. \end{align*} And by \eqref{e3.15} and \eqref{e3.5}, we have \begin{align*} V(t) & \geq(-4-8\delta)E(0) J(t)^{-1/\delta}\\ & \quad+4(1+\delta)[ (R_{u}+S_{u}+R_{v}+S_{v}) (T_{1}-t)(\| u_{0}\|_{2}^{2}+\| v_{0}\| _{2}^{2})+\Theta(t)] , \end{align*} where \begin{align*} \Theta(t) & =(R_{u}+S_{u}+R_{v}+S_{v})(P_{u} +Q_{u}+P_{v}+Q_{v})\\ & \quad-(\sqrt{R_{u}P_{u}}+\sqrt{Q_{u}S_{u}}+\sqrt{R_{v}P_{v}}+\sqrt {Q_{v}S_{v}})^{2}. \end{align*} By Schwarz inequality, $\Theta(t)$ is nonnegative. Hence, we have $$V(t)\geq(-4-8\delta)E(0)J( t)^{-1/\delta},\quad t\geq t_{0}.\label{e3.20}$$ Therefore, by \eqref{e3.16} and \eqref{e3.20}, we get $$\ \ J''^{1+\frac{1}{\delta}},\quad t\geq t_{0}.\label{e3.21}$$ Note that by Lemma \ref{lem3.3}, $J'(t)<0$ for $t>t_{0}$. Multiplying \eqref{e3.21} by $J'(t)$ and integrating it from $t_{0}$ to $t$, we get $J'^{2}\geq\alpha+\beta J(t) ^{2+\frac{1}{\delta}}\quad\text{for }t\geq t_{0},$ where \begin{gather} \alpha=\delta^{2}J(t_{0})^{2+\frac{2}{\delta}}\big[ (a'(t_{0})-\| u_{0}\|_{2}^{2}-\| v_{0} \|_{2}^{2})^{2}-8E(0)J(t_{0})^{\frac{-1}{\delta} }\big],\label{e3.22}\\ \beta=8\delta^{2}E(0).\label{e3.23} \end{gather} We observe that $\alpha>0\quad\text{if and only if}\quad E(0)<\frac{(a'(t_{0})-\| u_{0}\|_{2}^{2}-\| v_{0}\|_{2}^{2})^{2} }{8\big[ a(t_{0})+(T_{1}-t_{0})(\| u_{0}\| _{2}^{2}+\| v_{0}\|_{2}^{2})\big] }.$ Then by Lemma \ref{lem2.3}, there exists a finite time $T^{\ast}$ such that $\lim_{t\to T^{\ast-}}J(t)=0$ and the upper bound of $T^{\ast}$ is estimated respectively according to the sign of $E(0)$. This means that $$\lim_{t\to T^{\ast-}}\big\{ \int_{\Omega}( u^{2}+v^{2})\,dx+\int_{0}^{t}(\| u\|_{2}^{2}+\| v\|_{2}^{2})dt\big\} =\infty.\label{e3.24}$$ \begin{theorem} \label{thm3.4} Assume that {\rm (A1)} and that either one of the following statements is satisfied: \begin{itemize} \item[(1)] $E(0)<0$, \item[(ii)] $E(0)=0$ and $a'(0)>(\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2} ^{2})$ \item[(iii)] $00$ such that $\lim_{t\to T^{-}}\big[ \int_{\Omega}(|\nabla u| ^{2}+|\nabla v| ^{2})\,dx\big] =\infty.$ \begin{lemma} \label{lem3.5} For all $\lambda>1$, $\alpha\neq0$, there exists $\beta>0$ such that $$\xi^{4}+\alpha^{4}\eta^{4}\leq\lambda(\xi+\alpha\eta)^{4}+\beta\xi^{2}\eta ^{2},\quad \text{for all }\xi,\eta\in \mathbb{R}.\label{e3.25}$$ \end{lemma} \begin{proof} If $\eta=0$, then \eqref{e3.25} is true for $\lambda>1$, $\xi\in\mathbb{R}$. Now, let $x=\frac{\xi}{\eta}$, where $\xi,\eta\in\mathbb{R}$, and $\eta\neq0$. Then to show \eqref{e3.25} is equivalent to claim that for all $\lambda >1,\alpha\neq0$, there exists $\beta>0$ such that $h(x)\leq\beta x^{2}$, here $h(x)=x^{4}+\alpha^{4}-\lambda(x+\alpha)^{4}$, $x\in\mathbb{R}$. Since $h(x)$ is a continuous function, $h(0)=\alpha^{4}-\lambda\alpha^{4}<0$, and $h(\pm\infty)=-\infty$, there exists a finite number $M$ such that $M=\sup_{x\in\mathbb{R}}h(x)$. If $M\leq0$, we could choose any $\beta>0$. If $M>0$, since $h(0)<0$, there exists $\delta>0$ such that $h(x)<0$ for $|x| <\delta$. Thus, we could choose $\beta=M$ in this interval. For $|x| \geq\delta$, $h(x)\leq M=\frac{M}{\delta^{2}}\delta^{2}\leq\frac{M}{\delta ^{2}}x^{2}$. Therefore, from above discussion, we can take $\beta=\max \{\frac{M}{\delta^{2}},M\}$, and we have $h(x)\leq\beta x^{2}$, for $x\in\mathbb{R}$. \end{proof} \begin{theorem}[Nonexistence of global solutions] \label{thm3.6} If $10.\label{e3.30} Thus, from \eqref{e3.29}, \eqref{e3.30}, and$1-a''^{1-\alpha_{1}}is a positive number to be chosen later. From \eqref{e3.35}, we see $Z'(t)=k_{6}(1-\alpha_{1})(-E(t))^{-\alpha_{1}}(-E'(t))+a''(t),\quad t\geq0.$ By \eqref{e3.27}, \eqref{e3.33} and \eqref{e3.34}, we get \begin{aligned} Z'(t) & \geq\mu(-E(t))^{-\alpha_{1}}(-E'(t))+(-2E(t)) +2\int _{\Omega}(u_{t}^{2}+v_{t}^{2})\,dx \\ &\quad +\big[ 2-\varepsilon_{1}^{p+1}(-E(0))^{\frac{1}{4}-\frac{1}{p+1} } -\varepsilon_{2}^{q+1}(-E(0))^{\frac{1}{4}-\frac{1}{q+1}}\big] B(t), \end{aligned}\label{e3.36} where \begin{align*} \mu & =k_{1}(1-\alpha_{1})-c(\varepsilon_{1})^{-\frac{p+1}{p}}|\Omega |^{\frac{3-p}{4p}}(-E(0))^{\alpha_{1}+\frac{1}{4}-\frac{1}{p+1}}\\ & \quad-c(\varepsilon_{2})^{-\frac{q+1}{q}}|\Omega|^{\frac{3-q}{4q} }(-E(0))^{\alpha_{1}+\frac{1}{4}-\frac{1}{q+1}}. \end{align*} We choose $\varepsilon_{1}^{p+1}=\frac{1}{2}(-E(0))^{\frac{1}{p+1}-\frac{1}{4}} ,\quad\varepsilon_{2}^{q+1}=\frac{1}{2}(-E(0))^{\frac{1}{q+1}-\frac{1}{4}},$ andk_{1}$is sufficiently large such that$\mu>0$and$Z(0)>0$. Then \eqref{e3.36} becomes $$Z'(t)\geq\big[-2E(t)+\Vert u_{t}\Vert_{2}^{2}+\Vert v_{t}\Vert_{2} ^{2}+B(t)\big].\label{e3.37}$$ Hence$Z(t)>0$for$t\geq0$. Note that$r=1/(1-\alpha_{1})>1, from \eqref{e3.35}, and by Young's inequality and H\"{o}lder inequality, it follows that \begin{aligned} Z(t)^{r} & \leq2^{2(r-1)}\Big[ k_{1}^{r}(-E(t)) +\big|\int_{\Omega }u_{t}u\,dx\big| ^{r} +\big|\int_{\Omega}v_{t}v\,dx\big|^{r}\Big] \\ & \leq2^{2(r-1)}[ k_{1}^{r}(-E(t))+\|u_{t}\|_{2} ^{r}\|u\|_{2}^{r} +\|v_{t}\|_{2} ^{r}\|v\|_{2}^{r}]. \label{e3.38} \end{aligned} On the other hand, using H\"{o}lder inequality, we have $\Vert u_{t}\Vert_{2}^{r}\Vert u\Vert_{2}^{r}\leq c_{1}\Vert u_{t}\Vert_{2} ^{r}\Vert u\Vert_{4}^{r},$ herec_{1}=|\Omega|^{r/4}$. And by Young's inequality, we obtain $$\Vert u_{t}\Vert_{2}^{r}\Vert u\Vert_{2}^{r}\leq c_{2}(\Vert u_{t}\Vert _{2}^{r\beta_{1}}+\Vert u\Vert_{4}^{r\beta_{2}}),\label{e3.39}$$ where$\frac{1}{\beta_{1}}+\frac{1}{\beta_{2}}=1$,$c_{2}=c_{2}(c_{1}$,$\beta_{1}$,$\beta_{2})>0$. In particular, we take$r\beta_{1}=2$; that is,$\beta_{1}=2(1-\alpha_{1})$. Therefore, for$\alpha_{1}$small enough, the numbers$\beta_{1}$and$\beta_{2}$are close to 2. For$0<\alpha_{1} <\min\{\frac{1}{p+1}-\frac{1}{4},\frac{1}{q+1}-\frac{1}{4}\}, by \eqref{e3.25} and \eqref{e3.30}, we have \begin{align*} \Vert u\Vert_{4}^{r\beta_{2}}\big(\Vert u\Vert_{4}^{4}\big)^{r\beta_{2}/4} & \leq B(t)^{r\beta_{2}/4}\\ & =(\frac{1}{-E(0)}B(t))^{r\beta_{2}/4}(-E(0))^{r\beta_{2}/4}\\ & \leq c_{3}B(t) \end{align*} because $r\beta_{2}=\frac{2}{1-2\alpha_{1}}<4,$ wherec_{3}=(-E(0))^{\frac{r\beta_{2}}{4}-1}$. Then, by \eqref{e3.25}, we obtain $$\Vert u_{t}\Vert_{2}^{r}\Vert u\Vert_{2}^{r}\leq c_{4}\big(\Vert u_{t} \Vert_{2}^{2}+B(t)\big).\label{e3.40}$$ Similarly, we also get $$\Vert v_{t}\Vert_{2}^{r}\Vert v\Vert_{2}^{r}\leq c_{5}\big(\Vert v_{t} \Vert_{2}^{2}+B(t)\big),\label{e3.41}$$ here$c_{4}=c_{2}\max(1,\text{ }c_{3})$, and$c_{5}$is some positive constant. Then, from \eqref{e3.38}, \eqref{e3.40} and \eqref{e3.41}, we deduce that $$Z(t)^{r}\leq2^{2(r-1)}c_{6}[-2E(t)+\Vert u_{t}\Vert_{2}^{2}+\Vert v_{t} \Vert_{2}^{2}+B(t)],\label{e3.42}$$ where$c_{6}=\max\{\frac{k_{1}^{r}}{2},c_{4}+c_{5}\}$. Therefore, by \eqref{e3.37} and \eqref{e3.42}, we have $$Z'(t)\geq c_{7}Z(t)^{r},\label{e3.43}$$$c_{7}=\frac{1}{2^{2(r-1)}c_{6}}$. A simple integration of \eqref{e3.43} over$(0,t)$yields $$Z(t)\geq\big(Z(0)^{1-r}-c_{7}(r-1)t\big)^{-\frac{1}{\alpha_{1}-1} }.\label{e3.44}$$ Since$Z(0)>0$, \eqref{e3.44} shows that$Z$becomes infinite in a finite time$T\leq\frac{Z(0)^{1-r}}{c_{7}(r-1)}$. From \eqref{e3.1}, we have $$-2E(t)+\Vert u_{t}\Vert_{2}^{2}+\Vert v_{t}\Vert_{2}^{2}\leq 2B(t).\label{e3.45}$$ Thus, by \eqref{e3.37} and \eqref{e3.45}, we get $$Z(t)^{r}\leq3B(t).\label{e3.46}$$ By Poincar\'{e} inequality and H\"{o}lder inequality, we have $$B(t)\leq c_{8}(\Vert\nabla u\Vert_{2}+\Vert\nabla v\Vert_{2})^{4} ,\label{e3.47}$$$c_{8}=c_{8}(\alpha,\beta,\Omega)>0$. Hence, from \eqref{e3.46} and \eqref{e3.47}, we obtain $Z(t)^{r}\leq3c_{8}(\Vert\nabla u\Vert_{2}+\Vert\nabla v\Vert_{2})^{4}.$ Therefore, the proof is complete. \end{proof} \begin{thebibliography}{00} \bibitem {a1} M. Aassila; Global existence and global nonexistence of solutions to a wave equation with nonlinear damping and source terms, Asymptotic Analysis, 30(2002), 301-311. \bibitem {g1}V. Georgiev and D. Todorova; Existence of solutions of the wave equations with nonlinear damping and source terms, J. Diff. Eqns. 109(1994), 295-308. \bibitem {h1}M. Hosoya and Y. 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