\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 78, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/78\hfil Blow-up of solutions] {Blow-up of solutions to a coupled quasilinear viscoelastic wave system with nonlinear damping and source} \author[X. Zhang, S. Chai, J. Wu \hfil EJDE-2017/78\hfilneg] {Xiaoying Zhang, Shugen Chai, Jieqiong Wu} \address{Xiaoying Zhang \newline School of Mathmatical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China. \newline Department of Mathematics, Shanxi Agriculture University, Taigu, Shanxi 030800, China} \email{zxybetter@163.com} \address{Shugen Chai (corresponding author)\newline School of Mathmatical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China} \email{sgchai@sxu.edu.cn, Phone +86-351-7010555, Fax +86-351-7010979} \address{Jieqiong Wu \newline School of Mathmatical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China} \email{ jieqiong@sxu.edu.cn} \dedicatory{Communicated by Goong Chen} \thanks{Submitted February 26, 2016. Published March 21, 2017.} \subjclass[2010]{35A01, 35L53} \keywords{Blow up; quasilinear wave system; viscoelasticity} \begin{abstract} We study the blow-up of the solution to a quasilinear viscoelastic wave system coupled by nonlinear sources. The system is of homogeneous Dirichlet boundary condition. The nonlinear damping and source are added to the equations. We assume that the relaxation functions are non-negative non-increasing functions and the initial energy is negative. The competition relations among the nonlinear principal parts are not constant functions, the viscoelasticity terms, dampings and sources are analyzed by using perturbed energy method. The blow-up result is proved under some conditions on the nonlinear principal parts, viscoelasticity terms, dampings and sources by a contradiction argument. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} Let $\Omega$ be a bounded domain of $R^{n}(n\geq 1)$ with a smooth boundary $\partial \Omega$. Consider the following nonlinear viscoelastic system \begin{equation}\label{1.1} \begin{gathered} \begin{aligned} &|u_t|^{\rho}u_{tt}-\operatorname{div}(\rho_1(|\nabla u|^2)\nabla u) +\int_0^{t}g(t-\tau)\Delta u(x,\tau)d\tau+u_t+|u_t|^{m-1}u_t\\ &=f_1(u,v),\quad \Omega \times(0,T),\\ &|v_t|^{\rho}v_{tt}-\operatorname{div}(\rho_2(|\nabla v|^2)\nabla v) +\int_0^{t}h(t-\tau)\Delta v(x,\tau)d\tau+v_t+|v_t|^{r-1}v_t \\ &=f_2(u,v),\quad \Omega \times(0,T), \end{aligned}\\ u(x, t) =v(x,t)=0, \quad x\in \partial\Omega\times[0,T],\\ u(x, 0) = u_0(x), \quad u_t(x, 0) = u_1(x),\quad x\in \Omega,\\ v(x, 0) =v_0(x), \quad u_t(x, 0) =v_1(x),\quad x\in \Omega, \end{gathered} \end{equation} where $\rho>0$, $m,r>1$ and $\rho_1,\rho_2,f_1,f_2,g,h$ are functions satisfying the following assumptions: \begin{itemize} \item[(A1)] $\rho_{i}(s)=b_1+b_2s^{q_{i}}$ with $q_{i}\geq 0$ and $b_1, b_2>0$; $\rho_{i}(s)>0$, for $s>0.$ \item[(A2)] The relaxation functions $g$ and $h$ are of class $C^{1}$ and satisfy, for $s\geq 0$, \begin{gather*} g(s)\geq 0,\quad b_1-\int_0^{\infty}g(s)ds=l>0,\quad g'(s)\leq 0,\\ h(s)\geq 0,\quad b_1-\int_0^{\infty}h(s)ds=k>0,\quad h'(s)\leq 0. \end{gather*} \item[(A3)] Let $F(u,v)=a|u+v|^{p+1}+2b|uv|^{\frac{p+1}{2}}$ with $a,b>0$, $1
m$. Messaoudi \cite{m1} extended the results of \cite{g1} to the case that the initial energy is negative. Agre and Rammaha \cite{a1} extended the results of \cite{g1} by considering an initial-boundary value problem to the coupled wave equations. In the presence of the viscoelastic term, Messaoudi \cite{m2} considered the nonlinear viscoelastic equation \begin{equation}\label{1.3} u_{tt}-\Delta u+\int_0^{t}g(t-\tau)\Delta u(\tau)d\tau+au_t|u_t|^{m-1}=b|u|^{p-1}u, \quad \Omega \times(0,\infty), \end{equation} with initial conditions and Dirichlet boundary conditions. He proved that the weak solution with negative initial energy blew up if $p>m$ when $g$ satisfied some conditions. Messaoudi \cite{m3} considered the blow-up solution of \eqref{1.3} with $a=1$, $b=1$ and with small positive initial energy. Song \cite{s1} extended the results of \cite{m3} to the case that the initial energy is arbitrarily positive. For other related works on the viscoelastic wave equation, we refer the reader to \cite{c1,c3,z1}. Problem \eqref{1.1} with $\rho>0$ has also been extensively studied. Song \cite{s2} investigated the nonexistence of global solutions to the initial-boundary value problem of the following equation with positive initial energy \begin{equation}\label{1.4} |u_t|^{\rho}u_{tt}-\Delta u+\int_0^{t}g(t-\tau)\Delta u(\tau)d\tau+u_t|u_t|^{m-2} =|u|^{p-2}u,\quad \Omega \times(0,\infty). \end{equation} Liu \cite{l1} studied the general decay for the global solution and blow-up of solution to the equation \begin{equation}\label{1.5} |u_t|^{\rho}u_{tt}-\Delta u+\int_0^{t}g(t-\tau)\Delta u(\tau)d\tau -\Delta u_{tt}=|u|^{p-2}u,\quad \Omega \times(0,\infty). \end{equation} Cavalcanti et al.\ \cite{c2} studied the energy decay for the nonlinear viscoelastic problem \begin{equation}\label{1.6} |u_t|^{\rho}u_{tt}-\Delta u+\int_0^{t}g(t-\tau)\Delta u(\tau)d\tau-\Delta u_{tt} -\gamma \Delta u_t=0,\quad \Omega \times(0,\infty). \end{equation} A global existence result for $\gamma\geq 0$ as well as an exponential decay for $\gamma>0$ was established in \cite{c2}. When the source term $b|u|^{p-2}u$ appeared on the right side of system \eqref{1.6}, Messaoudi et al.\ \cite{m4} proved that the viscoelastic term was enough to ensure existence and uniform decay of global solutions provided that the initial data were in some stable set. For $\rho_{i}(s)=b_1+b_2s^{q_{i}}$ with $q_{i}\geq 0$ and $b_1, b_2>0$, Wu et al.\ \cite{w1} and \cite{w2} considered the blow-up of the initial boundary value problem (spatial dimension $n=1,2,3$) for the system \begin{equation}\label{1.7} \begin{gathered} u_{tt}-\operatorname{div}(\rho_1(|\nabla u|^2)\nabla u)+u_t +|u_t|^{m-1}u_t=f(u,v),\quad \Omega \times(0,T),\\ v_{tt}-\operatorname{div}(\rho_2(|\nabla v|^2)\nabla v)+v_t+|v_t|^{r-1}v_t=g(u,v), \quad \Omega \times(0,T). \end{gathered} \end{equation} For a single wave equation with $\rho_{i}(s)\geq b_1+b_2s^{q_{i}}$, $q_{i}\geq 0$, $b_1, b_2>0$, Hao et al.\ \cite{h1} studied the global existence and blow up of the solutions. We note that, in the literature mentioned above, only viscoelastic term was included in the equation or only nonlinear principal part (i.e. $\rho_i, i=1,2$, are not constant functions) was included. To the best of our knowledge, there are no papers considering the blow-up of the equation with both viscoelastic term and nonlinear principal part. The main goal of our paper is to prove that for $\rho_{i}(s)=b_1+b_2s^{q_{i}}$ the nonlinear coupled source terms still leads to blow-up of the solutions though there are viscoelastic terms in the equations. To be more precise, we prove that when $p>\max\{2q_1+1,2q_2+1\}$ and the relaxation functions satisfy that $\max\{\int_0^{\infty}g(s)ds,\int_0^{\infty}h(s)ds\}<\frac{q}{q+1}b_1$, the solutions of the system will blow up. Our method is borrowed partly from \cite{l1,w1}, but we must overcome some additional difficulty caused by the complex interaction among the nonlinear viscoelastic terms, the nonlinear principal parts, the coupled source terms and the nonlinear damping. \section{Preliminaries} In this section, we present some other assumptions and existence result of local solution. We use the following assumptions: \begin{itemize} \item[(A4)] $\rho>0$ if $n=1,2$ and $0<\rho<\frac{2}{n-2}$ if $n\geq 3$. \item[(A5)] $m
0$, and
\begin{gather*}
u\in L^{\infty}([0,T); W_0^{1,2q_1+2}(\Omega)\cap L^{p+1}(\Omega)), \\
v\in L ^{\infty}([0,T];W_0^{1,2q_2+2}(\Omega)\cap L^{p+1}(\Omega)),\\
u_t\in L^{\infty}([0,T); W_0^{1,2q_1+2}(\Omega)\cap L^{p+1}(\Omega)), \\
v_t\in L ^{\infty}([0,T];W_0^{1,2q_2+2}(\Omega)\cap L^{p+1}(\Omega))\\
u_{tt}\in L^{\infty}([0,T); L^2(\Omega)),\quad
v_{tt}\in L^{\infty}([0,T); L^2(\Omega))
\end{gather*}
\end{theorem}
Combining the arguments of \cite{g1,m3}, the following lemma can be
proved easily.
\begin{lemma}\label{lem2.1}
Let {\rm (A1)--(A4)} hold. And let $(u,v)$ be a solution of \eqref{1.1}.
Then $E(t)$ satisfies the inequality
\begin{equation} \label{2.2}
\begin{aligned}
E'(t) &= -\|u_t\|^2-\|u_t\|_{m+1}^{m+1}-\|v_t\|^2-\|v_t\|_{r+1}^{r+1}
+\frac{1}{2}(g'\circ\nabla u)(t) \\
&\quad +\frac{1}{2}(h'\circ\nabla v)(t) -\frac{1}{2}g(t)\|\nabla u\|^2
-\frac{1}{2}h(t)\|\nabla v\|^2\leq 0.
\end{aligned}
\end{equation}
\end{lemma}
\begin{lemma}[\cite{m1}] \label{lem2.2}
Suppose $p$ satisfies {\rm (A3)}. Then there exists a positive constant
$C(|\Omega|,p)$ such that
$$
\|u\|_{p+1}^{s}\leq C(|\Omega|,p)\Big(\|\nabla u\|^2+\|u\|_{p+1}^{p+1}\Big), \quad
\forall u\in H_0^{1}(\Omega),
$$
where $2\leq s\leq p+1$.
\end{lemma}
In this article, we use $\|\cdot\|$ and $\|\cdot\|_{p}$ denote the usual
$L^2(\Omega)$ norm and $L^{p}(\Omega)$ norm, respectively.
$B_1$ is the optimal constant of the Sobolev embedding
$H_0^{1}(\Omega)\hookrightarrow L^2(\Omega)$.
\section{Blow-up results}
In this section, we state and prove our main result.
\begin{theorem}\label{thm3.1}
Let {\rm (A1)--(A5)} hold. $q=\max\{q_1,q_2\}$. Assume the initial energy
$E(0)<0$ and
$$
\max\Big\{\int_0^{\infty}g(s)ds,\int_0^{\infty}h(s)ds\Big\}<\frac{q}{q+1}b_1,\quad
p>\max\{2q_1+1,2q_2+1\}.
$$
Then the solution of \eqref{1.1} blows up at finite time.
\end{theorem}
\begin{proof}
We use the contradiction method.
Suppose that the solution $(u,v)$ of \eqref{1.1} is global. Then
\begin{equation}\label{3.1}
\|u_t\|_{\rho+2}^{\rho+2}+\|\nabla u\|^2 +\|u\|_{p+1}^{p+1}
+\|v_t\|_{\rho+2}^{\rho+2}+\|\nabla v\|^2 +\|v\|_{p+1}^{p+1}\leq C,
\quad \forall t\geq 0.
\end{equation}
Set $M_1=\max_{t\in [0,T]}\|u\|_{p+1}^{p+1}$,
$M_2=\max_{t\in [0,T]}\|v\|_{p+1}^{p+1}$,
$M=M_1+M_2$. Let $H(t)=-E(t)$. Then by Lemma \ref{lem2.1}, the function
$H(t)$ is increasing. Moreover,
from $E(0)<0$ and (A3), we obtain
\begin{equation}\label{3.2}
\begin{aligned}
0 \frac{\varepsilon \gamma}{\widetilde{C}}L^{\frac{1}{1-\sigma}}(t), \quad
\text{for } t\geq T_0.
\end{equation}
The inequality above implies that $L(t)$ blows up at a finite time $T^*$ and
\begin{equation}\label{3.39}
T^{\ast}\leq\frac{\widetilde{C}(1-\sigma)}
{\varepsilon\gamma L^{\sigma/(1-\sigma)}(T_0)}.
\end{equation}
Furthermore, from \eqref{3.37} we obtain
\begin{equation}\label{3.40}
\lim_{t\to T^{\ast-}}\Big[\|u_t\|_{\rho+2}^{\rho+2}+\|\nabla u\|^2+\|u\|_{p+1}^{p+1}
+\|v_t\|_{\rho+2}^{\rho+2}+\|\nabla v\|^2+\|v\|_{p+1}^{p+1}\Big]=+\infty.
\end{equation}
If we choose the
$T>\frac{\widetilde{C}(1-\sigma)}{\varepsilon\gamma L^{\sigma/(1-\sigma)}(T_0)}$,
obviously, \eqref{3.40} contradicts \eqref{3.1}.
Thus, the solution of problem \eqref{1.1} blows up in finite time.
\end{proof}
\subsection*{Concluding remarks}
In this paper, we considered the blow-up of solutions to a coupled quasilinear
system with the nonlinear viscoelastic terms,
the nonlinear principal parts, the coupled source terms and the nonlinear dampings.
A sufficient condition under which the solutions of the system will blow up
at finite time is given. We show that the coupled sources are enough to
lead to the blow-up when the relaxation functions and the nonlinear principle
parts satisfy some conditions.
\subsection*{Acknowledgments}
This research supported by the National Natural Science
Foundation of China (11671240, 61403239, 61503230).
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\end{document}