\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:
\begin{equation}
u_{tt}-\Delta u(t)+a|u_{t}(t)|^{p-1}u_{t}(t)=f(u),\label{e1.7}
\end{equation}
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 $p<m$ with sufficiently negative initial energy, that is, in
the $L^{\infty}-$norm for suitable large initial data. Later, Ikehata
\cite{i1} showed that \eqref{e1.7} admits a global solution for sufficiently
small initial data for $p>1$. 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, $1<p$,
$q<3$, the nonexistence of global solutions is given in Theorem \ref{thm3.6}. Moreover,
estimates for the blow-up time $T$ are also given.

\section{Preliminary results}

In this section, we will give some lemmas and the local existence result in
Theorem \ref{thm2.4}.

\begin{lemma}[Sobolev-Poincar\'{e} inequality] \label{lem2.1}
If $2\leq p\leq6$,  then
\[
\|u\|_{p}\leq C(\Omega,p)\|\nabla u\|_{2},
\]
for $u\in H_{0}^{1}(\Omega)$, where
\[
C(\Omega,p)=\sup\big\{  \frac{\|u\|_{p}}{\|\nabla
u\|_{2}}: u\in H_{0}^{1}(\Omega),u\neq0\big\},
\]
and $\|\cdot\|_{p}$  denotes the norm of
$L^{p}(\Omega)$.
\end{lemma}


\begin{lemma}[\cite{l3}] \label{lem2.2}
Let $\delta>0$  and $B(t)\in C^{2}(0,\infty)$  be a nonnegative function
satisfying
\begin{equation}
B''(t)-4(\delta+1)B'(t)+4(\delta+1)B(t)\geq0.\label{e2.1}
\end{equation}
\textit{If }
\begin{equation}
B'(0)>r_{2}B(0)+K_{0},\label{e2.2}
\end{equation}
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
\begin{equation}
J'(t)^{2}\geq a+bJ(t)^{2+\frac{1}{\delta}},\quad \textit{for }t\geq
t_{0},\label{e2.3}
\end{equation}
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 $1<p,q<3$ in subsection 3.2. Let $(u,v)$ be a
solution of \eqref{e1.1}-\eqref{e1.6}, we define the energy functional
\begin{equation}
\begin{aligned} 
E(t) & =\frac{1}{2}\int_{\Omega}[|\nabla u| ^{2}
+|\nabla v| ^{2}+u_{t}^{2}+v_{t}^{2}+m_{1}^{2} u^{2}+m_{2}^{2}v^{2} \\ 
&\quad -2\lambda(u+\alpha v)^{4}-2\beta u^{2}v^{2}]\,dx,\quad \text{for }t\geq0.
 \end{aligned}\label{e3.1}
\end{equation}


\begin{lemma} \label{lem3.1}
$E(t)$  is a nonincreasing function for $t\geq 0$ and
\begin{equation}
\frac{d}{dt}E(t)=-\|u_{t}\|_{p+1}^{p+1}-\|
v_{t}\|_{q+1}^{q+1}.\label{e3.2}
\end{equation}
\end{lemma}


\begin{proof}
Multiplying \eqref{e1.1} by $u_{t}$ and \eqref{e1.2} by $v_{t}$, and
integrating them over $\Omega$. Then, adding them together, and integrating by
parts, we obtain
\[
E(t)-E(0)=-\int_{0}^{t}(\|u_{t}\|_{p+1} ^{p+1}+\|v_{t}\|_{q+1}^{q+1}
)dt\quad\text{for }t\geq0.
\]
Being the primitive of an integrable function, $E(t)$ is absolutely continuous
and equality \eqref{e3.2} is satisfied.
\end{proof}

\subsection{Case $p=q=1$}

In this subsection we consider \eqref{e1.1},\eqref{e1.2} with $p=q=1$:
\begin{gather}
\square u+u_{t}+m_{1}^{2}u =4\lambda(u+\alpha v)^{3}+2\beta uv^{2}
\quad\text{in }\Omega\times[0,T),\label{e3.3}\\
\square v+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{e3.4}
\end{gather}


Assumption:

\begin{itemize}
\item[(A1)] $m_{1}^{2}\xi^{2}+m_{2}^{2}\eta^{2}-2\lambda(\xi+\alpha\eta
)^{4}-2\beta\xi^{2}\eta^{2}<0$, for all $\xi$, $\eta\in\mathbb{R}$.
\end{itemize}

Definition: A solution $w(t)=(u(t),v(t))$ of \eqref{e3.3}, \eqref{e3.4}, and
\eqref{e1.3}-\eqref{e1.6} is called blow-up if there exists a finite time
$T^{\ast}$ such that
\[
\lim_{t\to T^{\ast-}}\big\{  \int_{\Omega}( u^{2}+v^{2})\,dx+\int_{0}^{t}(\|
u\|_{2}^{2}+\| v\|_{2}^{2})dt\big\}  =\infty.
\]
Let
\begin{equation}
a(t)=\int_{\Omega}(u^{2}+v^{2})\,dx+\int_{0}^{t}\int_{\Omega}(u^{2}
+v^{2})\,dxds,\quad\text{for }t\geq0.\label{e3.5}
\end{equation}


\begin{lemma} \label{lem3.2}
Assume {\rm (A1)}, and that
$0<\delta\leq 1/2$, then we have
\begin{equation}
\begin{aligned}
& a''(t)-4(\delta+1)\int_{\Omega}(u_{t}^{2}+v_{t}^{2})\,dx \\
& \geq(-4-8\delta)E(0)+(4+8\delta)
\int_{0}^{t}(\| u_{t}\|_{2}^{2}+\| v_{t}\|_{2}^{2})
dt.
\end{aligned} \label{e3.6}
\end{equation}
\end{lemma}


\begin{proof}
Form \eqref{e3.5}, we have
\begin{equation}
a'(t)=2\int_{\Omega}(uu_{t}+vv_{t}) \,dx+\| u\|_{2}^{2}+\| v\|_{2}
^{2}.\label{e3.7}
\end{equation}
By \eqref{e3.3}, \eqref{e3.4} and Divergence theorem, we get
\begin{equation}
\begin{aligned} a''(t)
& =2\int_{\Omega}(u_{t}^{2} +v_{t}^{2})\,dx
-2(\|\nabla u\|_{2} ^{2}+\|\nabla v\|_{2}^{2}+\|m_{1}u\| _{2}^{2}+\|m_{2}v\|_{2}^{2}) \\ 
&\quad +8\lambda\|u+\alpha v\|_{4}^{4}+8\beta\| uv\|_{2}^{2}. 
\end{aligned}\label{e3.8}
\end{equation}
By \eqref{e3.2}, we have from \eqref{e3.8}
\begin{align*}
&  a''(t)-4(\delta+1)\int_{\Omega}(u_{t}^{2}+v_{t}^{2})\,dx\\
&  =(-4-8\delta)E(0)+(4+8\delta) \int_{0}^{t}(\| u_{t}\|_{2}^{2}+\|
v_{t}\|_{2}^{2}) ds\\
& \quad+[ 4\delta(\|\nabla u\|_{2}^{2}+\|\nabla v\|_{2}^{2})+2(\|m_{1}
u\|_{2}^{2}+\| m_{2}v\|_{2}^{2})]\\
& \quad+(4\delta-2)[ \|m_{1}u\|_{2} ^{2}+\|m_{2}v\|_{2}^{2}-2\lambda\|u+\alpha
v\|_{4}^{4}-2\beta\|uv\|_{2}^{2}] .
\end{align*}
Therefore, from (A1), we obtain \eqref{e3.6}.
\end{proof}

We remark that (A1) is automatically true if $E(0)\leq0$.  Now, we consider
three different cases on the sign of the initial energy $E(0)$.

\noindent(1) If $E(0)<0$, then from \eqref{e3.6}, we have
\[
a'(t)\geq a'(0)-4(1+2\delta)E(0)t,\quad t\geq0.
\]
Thus we get $a'(t)>\| u_{0}\|_{2}^{2}+\|v_{0}\|_{2}^{2}$ for
$t>t^{\ast}$, where
\begin{equation}
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}
\end{equation}


\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
\begin{equation}
2\int_{0}^{t}\int_{\Omega}uu_{t}\,dx\,dt =\| u\|_{2}^{2}-\| u_{0}\|_{2}
^{2}.\label{e3.10}
\end{equation}
By H\"{o}lder inequality and Young's inequality, we have from \eqref{e3.10},
\begin{equation}
\| 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}
\end{equation}
Similarly,
\begin{equation}
\| 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}
\end{equation}
By H\"{o}lder inequality, Young$'$s inequality and then using
\eqref{e3.11} and \eqref{e3.12}, we have from \eqref{e3.7},
\begin{equation}
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}
\end{equation}
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
\begin{equation}
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}
\end{equation}
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
\begin{equation}
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}
\end{equation}
where $T_{1}>0$ is a certain constant which will be specified later. Then we
have
\begin{equation}
\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}
\end{equation}
where
\begin{equation}
\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}
\end{equation}
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{equation}
\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}
\end{equation}
By \eqref{e3.6}, we have
\begin{equation}
a''(t)\geq(-4-8\delta)E(0)+4(1+\delta)(R_{u}+S_{u}+R_{v}+S_{v})
.\label{e3.19}
\end{equation}
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
\begin{equation}
V(t)\geq(-4-8\delta)E(0)J( t)^{-1/\delta},\quad t\geq t_{0}.\label{e3.20}
\end{equation}
Therefore, by \eqref{e3.16} and \eqref{e3.20}, we get
\begin{equation}
\ \ J''^{1+\frac{1}{\delta}},\quad t\geq t_{0}.\label{e3.21}
\end{equation}
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
\begin{equation}
\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}
\end{equation}


\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)] $0<E(0)<\frac{(a'(t_{0})-\| u_{0}\|_{2}^{2}-\|
v_{0}\|_{2}^{2})^{2}}{8\big[  a(t_{0})+(T_{1}
-t_{0})(\| u_{0}\|_{2}^{2}+\| v_{0}\|_{2}^{2})
\big]  }$
and \eqref{e3.14} holds.
\end{itemize}
Then the solution $(u(t),v(t))$  blows up at finite time
$T^{\ast}$  in the sense of \eqref{e3.24}.
In case (i),
\[
T^{\ast}\leq t_{0}-\frac{J(t_{0})}{J'(t_{0})}.
\]
Furthermore, if $J(t_{0})<\min\{  1,\sqrt{-\alpha/\beta}\} $, we have
\[
T^{\ast}\leq t_{0}+\frac{1}{\sqrt{-\beta}}\ln\frac{\sqrt{-\alpha/\beta
}}{\sqrt{-\alpha/\beta}-J(t_{0})}.
\]
In case (ii),
\[
T^{\ast}\leq t_{0}-\frac{J(t_{0})}{J'(
t_{0})} \quad\text{or}\quad
T^{\ast}\leq t_{0}+\frac{J(t_{0})}{\sqrt{\alpha}}.
\]
In case (iii),
\[
T^{\ast}\leq\frac{J(t_{0})}{\sqrt{\alpha}}
\quad\text{or}\quad
T^{\ast}\leq t_{0}+2^{\frac{3\delta+1}{2\delta}}
\frac{\delta c}{\sqrt{\alpha}
}\big\{  1-[  1+cJ(t_{0})]  ^{\frac{-1}{2\delta}
}\big\},
\]
where $c=(\alpha/\beta)^{2+\frac{1}{\delta}}$,
here $\alpha$ and $\beta$ are given in
\eqref{e3.22}, \eqref{e3.23}.  Note that in case (i),
$t_{0}=t^{\ast}$ is given in \eqref{e3.9} and $t_{0}=0$ in case
(ii) and (iii).
\end{theorem}


We remark that the choice of $T_{1}$ in \eqref{e3.15} is possible under some
conditions as in \cite{w1,w2}.

\subsection{Case $1<p$, $q<3$}

In this subsection we consider \eqref{e1.1}, \eqref{e1.2}  with $1<p$, $q<3$:
\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),\\
\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).
\end{gather*}


\noindent\textbf{Definition:} A solution $(u,v)$ of \eqref{e1.1}-\eqref{e1.6}
is called blow-up if there exists a finite time $T>0$ 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
\begin{equation}
\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{equation}
\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 $1<p$, $q<3$, $E(0)<0$ and \eqref{e3.25}  holds,
then the solutions of \eqref{e1.1}-\eqref{e1.6} blow up at a finite
time $T$, $0<T\leq\frac{z(0)^{1-r}}{c_{7}(1-r)}$,
where $z(0)=k_{1}(-E(0))^{1-\alpha_{1}}+\int_{\Omega}(u_{1}u_{0}
+v_{1}v_{0})\,dx$,  here $k_{1}$, $\alpha_{1}$,  and
$r$ are certain positive constants given in the proof, and
$c_{7}$ is given in \eqref{e3.43}.
\end{theorem}


\begin{proof}
Let
\begin{equation}
a(t)=\frac{1}{2}\int_{\Omega}(u^{2}+v^{2})\,dx,\quad\text{for }t\geq
0.\label{e3.26}
\end{equation}
By differentiating, we obtain
\begin{gather*}
a'(t)=\int_{\Omega}(u_{t}u+v_{t}v)\,dx,\\
a''(t)=\int_{\Omega}(u_{t}^{2}+u_{tt}u+v_{t}^{2}+v_{tt}
v)\,dx,\quad\text{for }t\geq0.
\end{gather*}
By using \eqref{e1.1}, \eqref{e1.2} and \eqref{e3.2}, we obtain
\begin{equation}
\begin{aligned} a''(t)& =2\int_{\Omega}(u_{t}^{2}+v_{t}^{2})\,dx -2E(t)+2B(t)\\ 
&\quad -\int_{\Omega}|u_{t}| ^{p-1}u_{t}u\,dx-\int_{\Omega }|v_{t}| ^{q-1}v_{t}v\,dx,
 \end{aligned}\label{e3.27}
\end{equation}
where
\begin{equation}
B(t)=\lambda\Vert u+\alpha v\Vert_{4}^{4}+\beta\Vert uv\Vert_{2}
^{2}.\label{e3.28}
\end{equation}
By H\"{o}lder inequality, we observe that
\[
|\int_{\Omega}|u_{t}|^{p-1}u_{t}u\,dx|\leq|\Omega|^{\frac{3-p}{4(p+1)}}\Vert
u_{t}\Vert_{p+1}^{p}\Vert u\Vert_{4}.
\]
Then from \eqref{e3.25},
\begin{equation}
|\int_{\Omega}|u_{t}|^{p-1}u_{t}u\,dx|\leq|\Omega|^{\frac{3-p}{4(p+1)}}\Vert
u_{t}\Vert_{p+1}^{p}B(t)^{\frac{1}{4}}.\label{e3.29}
\end{equation}
Noting that from \eqref{e3.2} and \eqref{e3.1}, we have
\begin{equation}
B(t)\geq-E(t)\geq-E(0)>0.\label{e3.30}
\end{equation}
Thus, from \eqref{e3.29}, \eqref{e3.30}, and $1<p<3$, we obtain
\begin{equation}
\big|\int_{\Omega}|u_{t}|^{p-1}u_{t}u\,dx\big|\leq\Vert u_{t}\Vert_{p+1}
^{p}|\Omega|^{\frac{3-p}{4(p+1)}}B(t)^{\frac{1}{p+1}}(-E(t))^{\frac{1}
{4}-\frac{1}{p+1}}.\label{e3.31}
\end{equation}
Then, by Young's inequality,
\begin{equation}
\big|\int_{\Omega}|u_{t}|^{p-1}u_{t}u\,dx\big|\leq\left[  \varepsilon
_{1}^{p+1}B(t)+c(\varepsilon_{1})^{-\frac{p+1}{p}}|\Omega|^{\frac{3-p}{4p}
}\Vert u_{t}\Vert_{p+1}^{p+1}\right]  (-E(t))^{\frac{1}{4}-\frac{1}{p+1}
},\label{e3.32}
\end{equation}
here $\varepsilon_{1}$ is a positive constant to be specified later. Letting
$0<\alpha_{1}<\min\{\frac{1}{p+1}-\frac{1}{4},\frac{1}{q+1}-\frac{1}{4}\}$,
and by \eqref{e3.32} and \eqref{e3.30}, we have
\begin{equation}
\begin{aligned} |\int_{\Omega}|u_{t}| ^{p-1}u_{t}u\,dx| 
&\leq c(\varepsilon_{1})^{-\frac{p+1}{p}}|\Omega| ^{\frac{3-p}{4p}}
(-E(0))^{\alpha_{1}+\frac{1}{4}-\frac{1}{p+1}} (-E(t))^{-\alpha_{1}}(-E'(t)) \\ 
&\quad +\varepsilon_{1}^{p+1}B(t)(-E(0))^{\frac{1}{4}-\frac{1}{p+1}}. 
\end{aligned}\label{e3.33}
\end{equation}
In the same way, we have
\begin{equation}
\begin{aligned} \big|\int_{\Omega}|v_{t}| ^{q-1}v_{t}v\,dx\big| 
& \leq c(\varepsilon_{2})^{-\frac{q+1}{q}}
|\Omega| ^{\frac{3-q}{4q}}(-E(0))^{\alpha_{1}+\frac{1}{4}-\frac{1}{q+1}} 
(-E(t))^{-\alpha_{1}}(-E'(t)) \\ 
&\quad +\varepsilon_{2}^{q+1}B(t)(-E(0))^{\frac{1}{4}-\frac{1}{q+1}}, 
\end{aligned}\label{e3.34}
\end{equation}
here $\varepsilon_{2}$ is a positive constant. Now, we define
\begin{equation}
Z(t)=k_{1}(-E(t))^{1-\alpha_{1}}+a'(t),\quad t\geq0,\label{e3.35}
\end{equation}
where $k_{1}>-a''^{1-\alpha_{1}}$ 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{equation}
\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}
\end{equation}
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
\[
\varepsilon_{1}^{p+1}=\frac{1}{2}(-E(0))^{\frac{1}{p+1}-\frac{1}{4}}
,\quad\varepsilon_{2}^{q+1}=\frac{1}{2}(-E(0))^{\frac{1}{q+1}-\frac{1}{4}},
\]
and $k_{1}$ is sufficiently large such that $\mu>0$ and $Z(0)>0$. Then
\eqref{e3.36} becomes
\begin{equation}
Z'(t)\geq\big[-2E(t)+\Vert u_{t}\Vert_{2}^{2}+\Vert v_{t}\Vert_{2}
^{2}+B(t)\big].\label{e3.37}
\end{equation}
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{equation}
\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}
\end{equation}
On the other hand, using H\"{o}lder inequality, we have
\[
\Vert u_{t}\Vert_{2}^{r}\Vert u\Vert_{2}^{r}\leq c_{1}\Vert u_{t}\Vert_{2}
^{r}\Vert u\Vert_{4}^{r},
\]
here $c_{1}=|\Omega|^{r/4}$. And by Young's inequality, we obtain
\begin{equation}
\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}
\end{equation}
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
\[
r\beta_{2}=\frac{2}{1-2\alpha_{1}}<4,
\]
where $c_{3}=(-E(0))^{\frac{r\beta_{2}}{4}-1}$. Then, by \eqref{e3.25}, we
obtain
\begin{equation}
\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}
\end{equation}
Similarly, we also get
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
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}
\end{equation}
where $c_{6}=\max\{\frac{k_{1}^{r}}{2},c_{4}+c_{5}\}$. Therefore, by
\eqref{e3.37} and \eqref{e3.42}, we have
\begin{equation}
Z'(t)\geq c_{7}Z(t)^{r},\label{e3.43}
\end{equation}
$c_{7}=\frac{1}{2^{2(r-1)}c_{6}}$. A simple integration of \eqref{e3.43} over
$(0,t)$ yields
\begin{equation}
Z(t)\geq\big(Z(0)^{1-r}-c_{7}(r-1)t\big)^{-\frac{1}{\alpha_{1}-1}
}.\label{e3.44}
\end{equation}
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
\begin{equation}
-2E(t)+\Vert u_{t}\Vert_{2}^{2}+\Vert v_{t}\Vert_{2}^{2}\leq
2B(t).\label{e3.45}
\end{equation}
Thus, by \eqref{e3.37} and \eqref{e3.45}, we get
\begin{equation}
Z(t)^{r}\leq3B(t).\label{e3.46}
\end{equation}
By Poincar\'{e} inequality and H\"{o}lder inequality, we have
\begin{equation}
B(t)\leq c_{8}(\Vert\nabla u\Vert_{2}+\Vert\nabla v\Vert_{2})^{4}
,\label{e3.47}
\end{equation}
$c_{8}=c_{8}(\alpha,\beta,\Omega)>0$. 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}

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\end{document}