\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 92, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/92\hfil Blow-up results]
{Blow-up results for systems of nonlinear Klein-Gordon equations with
arbitrary positive initial energy}

\author[S.-T. Wu\hfil EJDE-2012/92\hfilneg]
{Shun-Tang Wu}

\address{Shun-Tang Wu \newline
General Education Center\\
National Taipei University of Technology\\
Taipei, 106 Taiwan}
\email{stwu@ntut.edu.tw}

\thanks{Submitted February 24, 2012. Published June 7, 2012.}
\subjclass[2000]{35L05, 35L15, 35L70}
\keywords{Blow-up; life span; damping; nonlinear
wave equations; \hfill\break\indent Klein-Gordon equation}

\begin{abstract}
 The initial boundary value problem for a system of nonlinear Klein-Gordon
 equations in a bounded domain is considered. We prove the  existence of
 local solutions by using a successive approximation method.
 Then, we show  blow-up results with arbitrary positive initial energy
 by a concavity method. Also estimates for the  lifespan of solutions are given.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In this article we study the existence and blow-up of local solutions for the
system of nonlinear Klein-Gordon equations 
\begin{equation}
(u_i)_{tt}-\Delta u_i+m_i^2u_i+(u_i)_t=f_i(u)\quad\text{in }
\Omega\times[0,T),\; i=1,2, \label{e1.1}
\end{equation}
with initial conditions
\begin{equation}
u(x,0)  =\phi(x)  ,\quad u_t(x,0)
=\varphi(x)  ,\quad x\in\Omega, \label{e1.2}
\end{equation}
and boundary conditions
\begin{equation}
u(x,t)  =0,\quad x\in\partial\Omega\times(0,T), \label{e1.3}
\end{equation}
where $u=(u_1,u_2)$, $\phi=(\phi_1,\phi_2)$,
$\varphi=(\varphi_1,\varphi_2) $, and $\Omega\subset R^N$,
$N\geq1$, is a bounded domain with smooth boundary $\partial\Omega$ so that
Divergence theorem can be applied and $T>0$. 
Let $\Delta=\sum_{j=1}^N\frac{\partial^2}{\partial x_{j}^2}$ be the Laplace operator,
$m_i\neq0$ is a real constant and $f_i(u)$ is a nonlinear function of $u$,
$i=1,2$.

Before stating our results, we first recall the existing results about the
initial boundary value problem for a single wave equation
\begin{equation}
u_{tt}-\Delta u+a| u_t| ^{m-1}u_t=b|u| ^{p-1}u,\label{e1.4}
\end{equation}
where $a>0$, $b>0$, $m\geq1$, and $p\geq1$. There are numerous results about
the global existence, asymptotic behavior and blow-up of solutions for \eqref{e1.4}.
Levine \cite{l1} firstly showed that the solutions with negative initial energy blow
up in finite time for equation \eqref{e1.4} with linear damping ($m=1$). Georgiev and
Todorova \cite{g2} extended Levine's result to nonlinear case ($m>1$). They showed
that solutions with negative initial energy continue to exist globally in time
if $m\geq p$ and blow-up in finite time if $p>m$ and the initial energy is
sufficiently negative. Later, Levine and Serrin \cite{l4} and Levine, Park, and
Serrin \cite{l3} generalized this result to an abstract setting and to unbounded
domains. By combining the arguments in \cite{g2} and \cite{l4}, 
Vitillaro \cite{v1} extended
these results to nonlinear damping ($m>1$) and the solution has positive
initial energy. Messaoudi \cite{m2} improved the work of \cite{g2} without imposing the
condition that energy is sufficiently negative. Similar results have also been
established by Todorova \cite{t1,t3} for different Cauchy problems. For related
results on a single wave equation, we refer the reader to \cite{m3,p1,z1}
 and the references therein.

On the other hand, Levine and Todorova \cite{l2} proved the local solution blows up
in finite time for some initial data with arbitrary high initial energy. Then
this result was improved by Todorova and Vitillaro \cite{t2}. However, they did not
give a sufficient condition for the initial data such that the corresponding
solutions blow up in finite time with arbitrary positive initial energy.
Recently, Wang \cite{w1} discussed the blow-up phenomena for equation \eqref{e1.4} with
$a=0$. They obtained a sufficient condition of the initial data such that the
solution of \eqref{e1.4} blows up in finite time when the positive initial energy is
arbitrarily large.

Now, we return to the initial boundary problem for the system of nonlinear
wave equations as follows
\begin{equation}
\begin{gathered}
(u_i)_{tt}-\Delta u_i+m_i^2u_i+| (
u_i)_t| ^{p_i-1}(u_i)_t
=f_i(u)\quad\text{in }\Omega\times[0,T),\; i=1,2,\\
u(x,0)  =\phi(x)  ,\quad u_t(x,0)
=\varphi(x)  ,\quad x\in\Omega,\\
u(x,t)  =0,\quad x\in\partial\Omega\times(0,T),
\end{gathered}\label{e1.5}
\end{equation}
where $p_1$, $p_2\geq1$ and $\Omega$ is a bounded domain with smooth
boundary. Reed \cite{r1} proposed this interesting problem without imposing
damping terms $| (u_i)_t| ^{p-1}(u_i)_t$ in \eqref{e1.5}
 to describe the interaction of scalar fields
$u_1$, $u_2$ of mass $m_1$, $m_2$ respectively. As in the case of a
single wave equation, it is worth noting that when the damping terms
$| (u_i)_t| ^{p_i-1}(
u_i)_t$ is absent, then the force term $f_i(u)$ causes finite
blow-up of solution for \eqref{e1.5}. In this direction,
 Wang \cite{w2} studied \eqref{e1.5} with
$f_1(u_1,u_2)=a_1| u_2| ^{q_2+1}|
u_1| ^{q_1-1}u_1$ and $f_2(u_1,u_2)=a_2|
u_1| ^{q_1+1}| u_2| ^{q_2-1}u_2$ and
obtained that the solutions blow up in finite time with arbitrary positive
initial energy. On the other hand, if the source term $f_i(u)$ is removed
from the equation, then the damping terms should assure global existence and
decay of solutions. However, when both damping and source terms are present,
then the analysis of their interaction and their influence on the behavior of
solutions becomes more difficult.  Agre and Rammaha \cite{a1}
 considered \eqref{e1.5} with
\begin{equation}
\begin{gathered}
f_1(u_1,u_2)=(r+1)\Big[  a| u_1+u_2|
^{r-1}(u_1+u_2)+b| u_1| ^{\frac{r-3}{2}}|
u_2| ^{\frac{r+1}{2}}u_1\Big]  ,\\
f_2(u_1,u_2)=(r+1)\Big[  a| u_1+u_2|
^{r-1}(u_1+u_2)+b| u_2| ^{\frac{r-3}{2}}|
u_1| ^{\frac{r+1}{2}}u_2\Big]  ,
\end{gathered}\label{e1.6}
\end{equation}
where $r\geq3$, $a>1$ and $b>0$. They showed the existence of global solutions
if $r\leq\min\{  p_1,p_2\}  $ and proved the blow-up of
solutions if $r>\min\{  p_1,p_2\}  $ and initial energy is
negative. Later, Alves et al \cite{a2} improved these results and they obtained
several results on the global, uniform decay rates, and blow up of solutions
in finite time when the initial energy is nonnegative by involving the Nehari
manifold. Recently, Li and Tsai \cite{l5} considered a class of nonlinear terms
which includes \eqref{e1.6} in a bounded domain where the global existence and
blow-up behavior of solutions \ without imposing damping terms were discussed.
However, on considering the blow-up properties, the initial energy can not be
arbitrarily large in that paper. This motivates us to consider the problem of
how to obtain the blow-up of solutions when the initial energy is arbitrarily large.

Inspired by these previous works \cite{l5,w1,w2}, in this present paper, we would
like to investigate the local existence and then establish a sufficient
condition of the initial data with arbitrarily high initial energy such that
the corresponding local solution of the system for the nonlinear Klein-Gordon
equations \eqref{e1.1}-\eqref{e1.3} blows up in finite time. The method used here are the
successive approximation method and the concavity method. In this way, we can
extend the result of \cite{w1} to a system with linear damping terms and the result
of \cite{l5} without setting any restriction on upper bound of the initial energy.
The paper is organized as follows. In section 2, we first introduced some
notations used throughout this paper and then state the local existence
Theorem \ref{thm2.4}. In section 3, we prove the main result
Theorem \ref{thm3.4} which shows
blow-up properties of solutions with highly positive initial energy.

\section{Exitance of local solutions}

In this section we shall discuss the existence of local solutions for
\eqref{e1.1}-\eqref{e1.3} by the method of successive approximations. 
First we give the notation which will be used throughout the paper. 
Let $W^{m,p}(\Omega)  $ be the usual Sobolev space. Specially, 
$W^{m,2}(\Omega)  $ and $W^{0,p}(\Omega)  $ will be marked by
$H^{m}(\Omega)  $ and $L^p(\Omega)  $,
respectively. And we denote $\| \cdot\|_p$ to be $L^p$-norm for
 $1\leq p\leq\infty$. $H_0^1(\Omega)$ is the closure of
$C_0^{\infty}(\Omega)$ with respect to the norm 
$\| u\|_{H_0^1}=\| \nabla u\|_2$.

Define
\begin{gather*}
H1=C^1([0,T];L^2(\Omega))\cap C^{0}([0,T];H_0^1
(\Omega)), \\
H2=C^2([0,T];L^2(\Omega))\cap C^1([0,T];H_0^1
(\Omega)),\text{ for }T>0.
\end{gather*}
Now, we make the following assumptions:
\begin{itemize}
\item[(A1)]
 $f_i :R^2\to \mathbb{R}$ is continuously differentiable such that for each
$u=(u_1,u_2)\in H_0^1(\Omega)\times H_0^1(\Omega)$, we have
$u_if_i\in L^1(\Omega)$, $i=1,2$ and $F(u)\in L^1(\Omega)$, where
\[
F(u)=\int_0^{u_1}f_1(s,u_2)ds+\int_0^{u_2}f_2(0,s)ds.
\]

\item[(A2)] $f_i(0)=0$ and for any $\rho>0$ there exists a constant
$k(\rho)>0$ such that
\[
\| f_i(u)-f_i(v)\|_2\leq k(\rho)\|u-v\|_{H_0^1\times H_0^1},\quad i=1,2,
\]
where $u,v\in H_0^1(\Omega)  \times H_0^1(
\Omega)  $ with $\| u\|_{H_0^1\times H_0^1
},\| v\|_{H_0^1\times H_0^1}$. 

\item[(A3)]
\[
\frac{\partial f_1}{\partial u_2}=\frac{\partial f_2}{\partial u_1}.
\]
\end{itemize}

Note that the function of the form $f_1(u_1,u_2)=u_1^{s-1}u_2
^{s}+u_1^p$, $f_2(u_1,u_2)=u_2^{s-1}u_1^{s}+u_2^{q}$ satisfy
the assumptions (A1)-(A3) where $1<s$, $p$, $q\leq\frac{N}{N-2}$ for
$N\geq3$ or $s$, $p$, $q>1$ for $N=1,2$.

\begin{lemma}[Sobolev-Poincar\'{e} \cite{m1}] \label{lem2.1}
 Let $2\leq p\leq\frac{2N}{N-2}$. then  the inequality 
\[
\| u\|_p\leq c_{s}\| \nabla u\|
_2,\text{\ \textit{for} }u\in H_0^1(\Omega),
\]
holds for some positive constant $c_{s}$.
\end{lemma}

\begin{lemma}[\cite{g1}] \label{lem2.2}
Let $\delta\geq0$, $T>0$ and 
$h$ be a Lipschitizan function over $[0,T)$.
Assume that $h(0)\geq0$ and $h'(t)+\delta h(t)>0$  for a.e.
$t\in(0,T)$. Then $h(t)>0$ for all $t\in(0,T)$.
\end{lemma}

Before proving the existence theorem for nonlinear equations
\eqref{e1.1}-\eqref{e1.3}, we
need the existence result for a linear wave equation which is given in
\cite{h1}.

\begin{lemma} \label{lem2.3}
Assume that $f\in W^{1,1}([0,T];L^2(\Omega)) $ and that 
$u_0\in H^2(\Omega)\cap H_0^1(\Omega)$ and 
$u_1\in H_0^1 (\Omega)$, then the linear problem with damping 
\begin{gather*}
u_{tt}-\Delta u+u_t    =f(t,x),\\
u(0)     =u_0,\quad u_t(0)  =u_1, \quad x\in\Omega,\\
u(x,t)    =0,\quad x\in\partial\Omega\times(0,T),
\end{gather*}
has a unique solution $u\in H2$.
\end{lemma}

\begin{theorem} \label{thm2.4}
Assume that the assumptions {\rm (A1)--(A3)}  hold and let
$(\phi_1,\phi_2)  \in H_0^1(\Omega)\times H_0^1(\Omega)$
 and $(\varphi_1,\varphi_2)  \in L^2(\Omega)\times L^2(\Omega)$.
Then  problem \eqref{e1.1}--\eqref{e1.3}
admits a unique solution $(u_{1,}u_2)  $ in
$H1\times H1$.
\end{theorem}

\begin{proof}
Since $H^2(\Omega)\cap H_0
^1(\Omega)$ is dense in $H_0^1(\Omega)$ and $H_0^1
(\Omega)$ is dense in $L^2(\Omega)  $, it suffices to consider
problem \eqref{e1.1}--\eqref{e1.3} for 
$\phi_i\in H^2(\Omega)\cap H_0^1(\Omega)$ and 
$\varphi_i\in H_0^1(\Omega)$, $i=1,2$. Let
$\{  u^{m}=(u_1^{m},u_2^{m})\}_{m\geq1}$ be a
sequence of solutions obtained by considering the  approximation
problem
\begin{equation} \label{e2.1}
\begin{gathered}
(u_i^{m+1})_{tt}-\Delta u_i^{m+1}+(u_i
^{m+1})_t    =-m_i^2u_i^{m}+f_i(u^{m}),\quad i=1,\quad 2,\\
u^{m+1}(x,0)   =\phi(x),\quad u_t^{m+1}(x,0)=\varphi(x),\quad
x\in\Omega,\\
u^{m+1}(x,t)    =0,\quad x\in\partial\Omega\times(0,T),
\end{gathered}
\end{equation}
with the initial function $u^1(x,0)=\phi(x)$.

Using Lemma \ref{lem2.3} and (A1)--(A2), we see that  \eqref{e2.1} has a
unique solution $u^{m}\in H2\times H2$. In the following, we would like to
estimate the solution obtained above. Multiplying by
 $(u_i^{m+1})_t$ on both sides of \eqref{e2.1} and then integrating it over
$\Omega$, we have
\begin{align*}
&  \int_{\Omega}(u_i^{m+1})_t[(u_i
^{m+1})_{tt}-\Delta u_i^{m+1}+(u_i^{m+1})_t]dx\\
&  =\int_{\Omega}(u_i^{m+1})_t[-m_i^2u_i
^{m}+f_i(u^{m})  ]dx.
\end{align*}
Using the Divergence theorem and H\"{o}lder inequality, we obtain
\begin{equation}
\frac{d}{dt}\| Du_i^{m+1}\|_2\leq\| m_i
^2u_i^{m}+f_i(u^{m})  \|_2, \label{e2.2}
\end{equation}
where $D\equiv(\partial_t,\nabla_{x})  $ and 
$\|Du_i\|_2^2=\int_{\Omega}(| (u_i)_t| ^2+| \nabla u_i|^2)  dx$. 
Integrating \eqref{e2.2} from 0 to $t$, we obtain
\begin{equation}
\| Du_i^{m+1}\|_2(t)\leq\| Du_i^{m+1}
\|_2(0)+\int_0^{t}\| m_i^2u_i^{m}+f_i(
u_1^{m},u_2^{m})  \|_2(r)dr. \label{e2.3}
\end{equation}
For simplicity, we denote
\begin{gather}
\beta_i    =\| Du_i^{m+1}\| (0)=(\|
\varphi_i\|_2^2+\| \phi_i\|_2
^2)  ^{1/2},\quad i=1,2,\nonumber \\
\beta   =\beta_1+\beta_2,\label{e2.4}\\
G_{m,i}    =m_i^2\| u_i^{m}\|_2+\|
f_i(u_1^{m},u_2^{m})  \|_2,\quad i=1,2,\; m\geq1,\label{e2.5}\\
H^{k}(t)    =\| Du^{k}\|_2(t)
=(\|Du_1^{k}\|_2+\| Du_2^{k}\|_2) (t),\quad k\geq1, \label{e2.6}
\end{gather}
where $Du^{k}=(Du_1^{k},Du_2^{k})$. Then using Lemma \ref{lem2.1}
and (A2), we have
\begin{equation}
G_{m,1}+G_{m,2}\leq c\| Du^{m}\|_2(t), \label{e2.7}
\end{equation}
here $c$ is some positive constant. It follows from \eqref{e2.3}-\eqref{e2.5}
 that
\begin{equation}
\| Du_i^2\|_2(t)   \leq\beta_i+\int_0^{t}
m_i^2\| \phi_i\|_2+\| f_i(\phi)  \|_2dt
 \leq\beta_i+G_{1,i}t. \label{e2.8}
\end{equation}
Thus by \eqref{e2.6} and \eqref{e2.8}, we obtain
\begin{equation}
H^2(t)\leq\beta+ct\| Du^1\|_2(t). \label{e2.9}
\end{equation}
Define
\begin{equation}
K_{\infty,\tau}(u^{i})  =\sup\{  \| Du^{i}
\|_2(t)\mid0\leq t\leq\tau\}  , \label{e2.10}
\end{equation}
and take a constant $M>\beta$. Then $H^1(t)\leq M$, and hence $K_{\infty
,\tau}(u^1)  \leq M$. Therefore, from \eqref{e2.9}, we see that
\[
H^2(t)\leq\beta+ctM\leq M,
\]
provided that $\tau=(M-\beta)/(cM)$. That is,
 $K_{\infty,\tau}(u^2)  \leq M$. Suppose that
$K_{\infty,\tau}(u^{m})  \leq M$, then,
 using \eqref{e2.3}, \eqref{e2.5}, \eqref{e2.7} and \eqref{e2.10}, we obtain
\begin{equation}
\begin{aligned}
H^{m+1}(t)
&  \leq\beta+\int_0^{t}(G_{m,1}+G_{m,2})(r)dr\\
&  \leq\beta+\int_0^{t}c\| Du^{m}\|_2(r)dr\\
&  \leq\beta+cK_{\infty,\tau}(u^{m})  t\leq M,\quad 0\leq
t\leq\tau.
\end{aligned}\label{e2.12}
\end{equation}
Thus $K_{\infty,\tau}(u^{m+1})  \leq M$. Hence, we have
\begin{equation}
K_{\infty,\tau}(u^{m})  \leq M,\text{ for all }m\geq1. \label{e2.13}
\end{equation}
Below we shall show that $\{  u^{m}\}_{m\geq1}$ is a Cauchy
sequence in $H1\times H1$. Let $z^{m}=u^{m+1}-u^{m}$. From \eqref{e2.1}, for $i=1,2$,
we have
\begin{equation} \label{e2.14}
\begin{gathered}
(z_i^{m})_{tt}-\Delta z_i^{m}+(z_i^{m})
_t    =-m_i^2z_i^{m-1}+f_i(u^{m})  -f_i(
u^{m-1})  ,\\
z^{m}(x,0)    =0,\quad z_t^{m}(x,0)=0,\quad x\in\Omega,\\
z^{m}(x,t)    =0,\quad x\in\partial\Omega\times(0,T).
\end{gathered}
\end{equation}
As in the previous arguments, we obtain
\begin{equation}
\begin{split}
&\| Dz^{m}\|_2(t)\\
&\leq\| Dz^{m}\|_2(0)+\sum_{i=1}^2
\int_0^{t}(m_i^2\| z_i^{m-1}\|_2+\|
f_i(u^{m})  -f_i(u^{m-1})  \|_2)dr.
\end{split}\label{e2.15}
\end{equation}
From \eqref{e2.14}, we obtain $\| Dz^{m}\|_2(0)=0$.
 Then, by \eqref{e2.13}, Lemma \ref{lem2.1} and (A2), we have
\[
\| Dz^{m}\|_2(t)\leq L\int_0^{t}\|
Dz^{m-1}\|_2(r)dr,\quad 0\leq t\leq\tau,
\]
where $L$ is a constant depending on $m_1$, $m_2$ and Sobolev constant.
Thus by induction, we obtain
\begin{equation}
K_{\infty,\tau}(z^{m})  \leq L\tau K_{\infty,\tau}(
z^{m-1})  \leq\cdots\leq(L\tau)  ^{m-1}K_{\infty,\tau
}(z^1)  . \label{e2.16}
\end{equation}
Therefore, for any positive integer $p$ and $L\tau\in(0,1)$, we see that
\begin{align*}
K_{\infty,\tau}(u^{m+p}-u^{m})
&  \leq((L\tau)  ^{m+p-2}+\cdots+(L\tau)  ^{m-1})
K_{\infty,\tau}(u^2-u^1) \\
&  \leq\frac{(L\tau)  ^{m-1}}{1-L\tau}K_{\infty,\tau}(
u^2-u^1)  \to0 \quad\text{ as }m\to\infty.
\end{align*}
Hence, the Cauchy sequence $\{  u^{m}\}_{m\geq1}$ converges in
$H1\times H1$ and the limit function $u=\lim_{m\to\infty}u^{m}$
in $H1\times H1$ is a solution defined on $[0,\tau)$ for problem
\eqref{e1.1}--\eqref{e1.3}.

\noindent\textbf{Uniqueness.} 
Let $u$ and $\widehat{u}$ be two
solutions defined on $[0,T)$ of problem \eqref{e1.1}-\eqref{e1.3}. 
Set $w=u-\widehat{u}$.
From \eqref{e1.1}, we have
\begin{gather*}
(w_i)_{tt}-\Delta w_i+(w_i)_t  =-m_i^2w_i+f_i(u)  -f_i(\widehat{u})
,\quad i=1, 2\\
w(x,0)    =0,\quad w_t(x,0)=0,\quad x\in\Omega,\\
w(x,t)    =0,\quad x\in\partial\Omega\times(0,T).
\end{gather*}
Similar to \eqref{e2.15}, we obtain
\[
\| Dw\|_2^2(t)\leq\| Dw\|_2 ^2(0)+c\int_0^{t}\| Dw\|_2^2(r)dr.
\]
The Gronwall's inequality implies
\[
\| Dw\|_2^2(t)=0,\text{ for }0\leq t<T.
\]
Therefore, we have $u=\widehat{u}$. 
\end{proof}

\section{Blow-up property}

In this section, we shall investigate blow-up phenomena of solutions of
system \eqref{e1.1}-\eqref{e1.3} with $m_1=m_2=1$. For this purpose, we further make
the following assumption:
\begin{itemize}
\item[(A4)] 
there exists a positive constant $\delta>0$ such that
\[
u_1f_1(u)+u_2f_2(u)\geq(2+4\delta)F(u),\quad \text{for all }u_1,u_2\in \mathbb{R},
\]
where $F(u)$ is given in (A1).
\end{itemize}

\noindent\textbf{Definition.} 
A solution $(u_1(t),u_2(t)) $ of \eqref{e1.1}-\eqref{e1.3} is said to blow up if
there exists a finite time $T$ such that
\begin{equation}
\lim_{t\to T^{-}}(\Vert u_1(t)\Vert_2^2+\Vert u_2(t)\Vert_2^2)  =\infty. \label{e3.1}
\end{equation}

Let $(u_1(t),u_2(t))  $ be the solution of \eqref{e1.1}-\eqref{e1.3}, we
define the energy function
\begin{equation}
E(t)=\frac{1}{2} \sum_{i=1}^2 [\| (u_i)_t\|_2^2+\Vert\nabla
u_i\Vert_2^2+\Vert u_i\Vert_2^2]-\int_{\Omega}F(u)dx,\quad
t\geq0 \label{e3.2}
\end{equation}
and
\begin{equation}
I(u(t))\equiv I(t)=
\sum_{i=1}^2 [\Vert\nabla u_i\Vert_2^2+\Vert u_i\Vert_2^2]-\int_{\Omega}
\sum_{i=1}^2 u_if_i(u)dx. \label{e3.3}
\end{equation}

\begin{lemma} \label{lem3.1}
 Let $u$ be a solution of \eqref{e1.1}-\eqref{e1.3}.
Then $E(t)$ is a nonincreasing function and 
\begin{equation}
E(t)  =E(0)  -\int_0^{t}
\sum_{i=1}^2 \| (u_i)_t\|_2^2dt. \label{e3.4}
\end{equation}
\end{lemma}

\begin{proof} 
By differentiating \eqref{e3.2} and using \eqref{e1.1}-\eqref{e1.3}, (A1) and
(A3), we obtain
\[
\frac{dE(t)}{dt}=- \sum_{i=1}^2 \| (u_i)_t\|_2^2.
\]
Thus, the result of Lemma \ref{lem3.1} follows.
\end{proof}

 \begin{lemma} \label{lem3.2}
Assume {\rm (A4)} 
and that $(\phi_1,\varphi_1), (\phi_2,\varphi_2)\in H_0^1(\Omega)\times L^2
(\Omega)$ satisfy  $E(0)>0$, $I(0)<0$,
\begin{gather}
\Vert\phi_1\Vert_2^2+\Vert\phi_2\Vert_2^2    >\frac{1+2\delta
}{\delta}E(0),\label{e3.5}\\
\int_{\Omega}(\phi_1\varphi_1+\phi_2\varphi_2)  dx  
>0. \label{e3.6}
\end{gather}
Then 
\[
\Vert u_1(t)\Vert_2^2+\Vert u_2(t)\Vert_2^2>\frac{1+2\delta
}{\delta}E(0)\quad \text{and}\quad I(t)<0,
\]
for all $t\in[0,T)$.
\end{lemma}

\begin{proof} First, we prove that $I(t)<0$, for all $t\in[0,T)$. 
Suppose not, then there exists $T^{\ast}>0$ such that 
$T^{\ast}=\min\{  t\in[0,T);I(t)=0\}  $. We
define
\[
G(t)=\int_{\Omega}(u_1^2(x,t)+u_2^2(x,t))  dx.
\]
Using \eqref{e1.1}, we have
\begin{gather*}
G'(t)=2\int_{\Omega} \sum_{i=1}^2 u_i(u_i)_tdx, \\
G''(t)=2\int_{\Omega} \sum_{i=1}^2 ((u_i)_t^2-| \nabla u_i|
^2-u_i^2+u_if_i(u))  dx-2\int_{\Omega} \sum_{i=1}^2
u_i(u_i)_tdx.
\end{gather*}
Then,  from \eqref{e3.3} it follows that
\begin{equation}
G''(t)+G'(t)=2[ \sum_{i=1}^2 \int_{\Omega}(u_i)_t^2dx-I(t)]>0, \label{e3.7}
\end{equation}
for all $t\in[0,T^{\ast})$. By Lemma \ref{lem2.2} and \eqref{e3.6}, we obtain
$G'(t)>0$, for all $t\in[0,T^{\ast})$. This implies $G(t)$ is
strictly increasing on $[0,T^{\ast})$. Thus, from \eqref{e3.5}, we have
\[
G(t)>G(0)>\frac{1+2\delta}{\delta}E(0),
\]
for all $t\in(0,T^{\ast})$. From the continuity of $u(t)$ at $t=T^{\ast}$, we
see that
\begin{equation}
G(T^{\ast})=\sum_{i=1}^2 \| u_i(T^{\ast})\|_2^2>\frac{1+2\delta}{\delta}E(0).
\label{e3.8}
\end{equation}
On the other hand, from \eqref{e3.2} and Lemma \ref{lem3.1}, we have
\begin{equation}
\begin{split}
&\sum_{i=1}^2 \big(\Vert\nabla u_i(T^{\ast})\Vert_2^2+\| u_i(T^{\ast})\|_2^2\big)  
-2\int_{\Omega}F(u_1(T^{\ast}),u_2(T^{\ast}))dx\\
&\leq 2E(T^{\ast})\leq2E(0).
\end{split} \label{e3.9}
\end{equation}
Noting that from the assumption $I(T^{\ast})=0$ and (A4) give us
\begin{equation}
\sum_{i=1}^2 \big(\Vert\nabla u_i(T^{\ast})\Vert_2^2+\| u_i(T^{\ast})\|_2^2\big)  
\geq (2+4\delta)\int_{\Omega}F(u_1(T^{\ast}),u_2(T^{\ast}))dx, \label{e3.10}
\end{equation}
which together with \eqref{e3.9} implies
\[
\sum_{i=1}^2
\big(\Vert\nabla u_i(T^{\ast})\Vert_2^2+\| u_i(T^{\ast
})\|_2^2\big)  \leq\frac{1+2\delta}{\delta}E(0).
\]
It is a contradiction to \eqref{e3.8}. Hence, $I(t)<0$, for all $t\in[0,T)$.
Therefore, following the same arguments as above, we deduce that $G(t)$ is
strictly increasing on $[0,T)$ and
\[
\Vert u_1(t)\Vert_2^2+\Vert u_2(t)\Vert_2^2>\frac{1+2\delta}{\delta}E(0),
\]
for all $t\in[0,T)$.
\end{proof}

Now, let
\begin{equation}
a(t)  =\sum_{i=1}^2
\Big(\int_{\Omega}u_i^2dx+\int_0^{t}\Vert u_i\Vert_2
^2dt\Big)  ,\quad t\geq0. \label{e3.11}
\end{equation}
We need the following lemma to derive our result.

\begin{lemma} \label{lem3.3}.
Assume that {\rm (A1), (A3)  (A4)} hold. Then 
\begin{equation}
a''(t)  \geq4(\delta+1)  \int_{\Omega}
\sum_{i=1}^2
(u_i)_t^2dx+(4+8\delta)  \int_0^{t}
\sum_{i=1}^2 \Vert(u_i)_t\Vert_2^2dt. \label{e3.12}
\end{equation}
\end{lemma}

\begin{proof}
 Form \eqref{e3.11} and using  \eqref{e1.1}, we have
\begin{equation}
a'(t)  =\sum_{i=1}^2
\Big(\int_{\Omega}2u_i(u_i)_tdx+\Vert u_i\Vert
_2^2\Big)  , \label{e3.13}
\end{equation}
and
\begin{equation}
a''(t) =2 \sum_{i=1}^2
\Big(\int_{\Omega}(u_i)_t^2dx-\| \nabla u_i\|_2^2-\| u_i\|_2^2\Big)
+2\int_{\Omega} \sum_{i=1}^2 u_if_i(u)dx. \label{e3.14}
\end{equation}
Employing \eqref{e3.2} , \eqref{e3.4} and (A4), we obtain
\begin{align*}
a''(t)   &  =4\int_{\Omega} \sum_{i=1}^2 (u_i)_t^2dx-4E(t)+2\int_{\Omega}(u_1
f_1(u)+u_2f_2(u)-2F(u))  dx\\
& \geq 4\int_{\Omega} \sum_{i=1}^2 (u_i)_t^2dx-4E(0)+4\int_0^{t}
\sum_{i=1}^2 \Vert(u_i)_t\Vert_2^2dt+8\delta\int_{\Omega}F(u)dx.
\end{align*}
Then, using \eqref{e3.2} and \eqref{e3.4} again, we see that
\begin{align*}
a''(t)   &  \geq4(1+\delta)\int_{\Omega}
\sum_{i=1}^2 (u_i)_t^2dx+4\delta
\sum_{i=1}^2 \Vert\nabla u_i\Vert_2^2+4\delta(
\sum_{i=1}^2 \Vert u_i\Vert_2^2-\frac{1+2\delta}{\delta}E(0)) \\
&  +4(1+2\delta)\int_0^{t}
\sum_{i=1}^2 \Vert(u_i)_t\Vert_2^2dt.
\end{align*}
Therefore, from Lemma \ref{lem3.2}, we obtain \eqref{e3.12}.
\end{proof}

Now, we are in a position to state and prove our main result.

\begin{theorem} \label{thm3.4} 
Assume that {\rm (A1)-(A4)}  hold. Also assume that
$(\phi_1,\varphi_1)$, $(\phi_2,\varphi_2)\in H_0
^1(\Omega)\times L^2(\Omega)$ satisfy the assumptions of Lemma
\ref{lem3.2}. Then the local solution $(u_1(t)$, $u_2(t))$  of
\eqref{e1.1}-\eqref{e1.3}  blows up at finite time 
$T^{\ast}$ in the sense of \eqref{e3.1}.
Moreover, if 
\[
2\delta\int_{\Omega}(\phi_1\varphi_1+\phi_2\varphi_2)
dx>\| \phi_1\|_2^2+\| \phi_2\|_2^2,
\]
then the finite time $T^{\ast}$  is estimated by
\begin{equation}
T^{\ast}\leq\frac{\| \phi_1\|_2^2+\| \phi
_2\|_2^2}{2\delta\int_{\Omega}(\phi_1\varphi_1
+\phi_2\varphi_2)  dx-(\| \phi_1\|
_2^2+\| \phi_2\|_2^2)  }.\label{e3.15}
\end{equation}
\end{theorem}

\begin{proof}
 We first note that
\begin{equation}
2\int_0^{t}\int_{\Omega}u_i(u_i)_t\,dx\,dt=\Vert u_i
\Vert_2^2-\Vert\phi_i\Vert_2^2.\label{e3.16}
\end{equation}
By H\"{o}lder inequality and Young's inequality,from
\eqref{e3.16} we have 
\begin{equation}
\Vert u_i\Vert_2^2\leq\Vert\phi_i\Vert_2^2+\int_0^{t}\Vert
u_i\Vert_2^2dt+\int_0^{t}\Vert(u_i)_t\Vert_2
^2dt,\quad i=1,2.\label{e3.17}
\end{equation}
Next, we will find the estimate for the life span of $a(t)$.
Let
\begin{equation}
J(t)  =\big[a(t)  +(T_1-t)
\sum_{i=1}^2 \Vert\phi_i\Vert_2^2\big]^{-\delta},\quad \text{for }t\in[0,T_1],
\label{e3.18}
\end{equation}
where $T_1>0$ is a certain constant which will be specified later. Then we
have
\begin{gather}
J'(t)  =-\delta J(t)  ^{1+\frac{1}{\delta}}(a'(t)  -
\sum_{i=1}^2 \Vert\phi_i\Vert_2^2)  , \label{e3.19}\\
J''(t)  =-\delta J(t)  ^{1+\frac
{2}{\delta}}V(t)  , \label{e3.20}
\end{gather}
where
\begin{equation}
V(t)  =a''(t)  \big[a(t)  +(T_1-t)
\sum_{i=1}^2 \Vert\phi_i\Vert_2^2\big]-(1+\delta)  
\Big(a'(t)  - \sum_{i=1}^2 \Vert\phi_i\Vert_2^2\Big)  ^2. \label{e3.21}
\end{equation}
For simplicity of calculation,  for $i=1,2$, we denote
\[
P_i    =\int_{\Omega}u_i^2dx,\quad Q_i=\int_0^{t}\|
u_i\|_2^2dt,\quad 
R_i   =\int_{\Omega}(u_i)_t^2dx,\quad S_i
=\int_0^{t}\| (u_i)_t\|_2^2dt.
\]
From \eqref{e3.13}   \eqref{e3.16}, and H\"{o}lder inequality, we obtain
\begin{equation} \label{e3.22}
\begin{aligned}
a'(t) & = \sum_{i=1}^2 \Big(\int_{\Omega}2u_i(u_i)_tdx+\Vert\phi_i\Vert
_2^2\Big)  +2 \sum_{i=1}^2 \int_0^{t} \int_{\Omega} u_i(u_i)_t \,dx\,dt\\
&  \leq2(\sqrt{R_1P_1}+\sqrt{Q_1S_1}+\sqrt{R_2P_2}+\sqrt
{Q_2S_2})+ \sum_{i=1}^2
\Vert\phi_i\Vert_2^2.
\end{aligned}
\end{equation}
By \eqref{e3.12}, we have
\begin{equation}
a''(t)  \geq4(1+\delta)  (R_1+S_1+R_2+S_2)  . \label{e3.23}
\end{equation}
Thus, from \eqref{e3.22}, \eqref{e3.23}, \eqref{e3.21} and \eqref{e3.18}, we obtain
\begin{align*}
V(t)   &  \geq[4(1+\delta)  (R_1+S_1+R_2+S_2)  ]J(t)  ^{-1/\delta}\\
&\quad  -4(1+\delta)  (\sqrt{R_1P_1}+\sqrt{Q_1S_1}+\sqrt{R_2P_2}+\sqrt{Q_2S_2})  ^2.
\end{align*}
Further, by \eqref{e3.18}$ and \eqref{e3.11}$, we deduce that
\[
V(t)  \geq4(1+\delta)  \big[(R_1 +S_1+R_2+S_2)  (T_1-t) \sum_{i=1}^2
\Vert\phi_i\Vert_2^2+\Theta(t)\big],
\]
where
\begin{align*}
\Theta(t)  &  =(R_1+S_1+R_2+S_2)  (P_1
+Q_1+P_2+Q_2) \\
& \quad -(\sqrt{R_1P_1}+\sqrt{Q_1S_1}+\sqrt{R_2P_2}+\sqrt {Q_2S_2})  ^2.
\end{align*}
By Schwartz inequality, $\Theta(t)$ is nonnegative. Hence, we have
\begin{equation}
V(t)  \geq0,\quad \text{for }t\geq0. \label{e3.24}
\end{equation}
Therefore by \eqref{e3.20} and \eqref{e3.24}, we obtain
$J''(t)  \leq0$ for $t\geq0$, and then
\begin{equation}
J(t)\leq J(0)+J'(0)t, \quad \text{for }t\geq0.\label{e3.25}
\end{equation}
Also, we note that
\[
J(0)>0\quad\text{and}\quad J'(0)<0
\]
due to \eqref{e3.18}, \eqref{e3.19} and \eqref{e3.6}. Hence, if we choose
$T_1\geq-J(0)/J'(0)$, from \eqref{e3.25}, there exists a finite time
$T^{\ast}\leq T_1$ such that
\[
\lim_{t\to T^{\ast-}}J(t)  =0.
\]
Then, it follows from the definition on $J(t)$ by \eqref{e3.18} that
\[
\lim_{t\to T^{\ast-}}
\sum_{i=1}^2 \Big(\Vert u_i\Vert_2^2+\int_0^{t}\Vert u_i(s)\Vert_2
^2ds\Big)  =\infty,
\]
which implies that
\[
\lim_{t\to T^{\ast-}} \sum_{i=1}^2 \Vert u_i\Vert_2^2=\infty.
\]
Moreover, if
\[
2\delta\int_{\Omega}(\phi_1\varphi_1+\phi_2\varphi_2)
dx>\| \phi_1\|_2^2+\| \phi_2\|_2^2,
\]
the upper bound $T^{\ast}$ can be estimated as
\[
T^{\ast}\leq\frac{\| \phi_1\|_2^2+\| \phi
_2\|_2^2}{2\delta\int_{\Omega}(\phi_1\varphi_1
+\phi_2\varphi_2)  dx-(\| \phi_1\|
_2^2+\| \phi_2\|_2^2)  }.
\]
This completes the proof.
\end{proof}

\begin{example} \label{examp3.5} \rm
Consider the system \eqref{e1.1}-\eqref{e1.3} with
\[
f_1(u_1,u_2)=u_1^2u_2,\quad f_2(u_1,u_2)=u_1u_2^2;
\]
that is, we consider the  problem
\begin{equation}
\begin{gathered}
(u_1)_{tt}-\Delta u_1+u_1+(u_1)
_t=u_1^2u_2\quad \text{in }\Omega\times[0,T),\\
(u_2)_{tt}-\Delta u_2+u_2+(u_2)
_t=u_2^2u_1\quad \text{in }\Omega\times[0,T),\\
u_1(x,0)=\phi_1,\quad u_2(x,0)=\phi_2,\quad x\in\Omega,\\
(u_1)_t(x,0)=\varphi_1,\quad (u_2)_t(x,0)=\varphi_2,\quad x\in\Omega,\\
u_1(x,t)=0,\quad u_2(x,t)=0,\quad x\in\partial\Omega\times(0,T).
\end{gathered}
\label{e3.26}
\end{equation}
By \eqref{e3.2} and \eqref{e3.3}, we have
\begin{gather*}
E(t)  =\frac{1}{2}\sum_{i=1}^2 \big[\| (u_i)_t\|_2^2+\Vert\nabla
u_i\Vert_2^2+\Vert u_i\Vert_2^2\big]-\frac{1}{2}\| u_1
^2u_2^2\|_2^2, \\
I(t)   =\sum_{i=1}^2 \big[\Vert\nabla u_i\Vert_2^2+\Vert u_i\Vert_2^2\big]
-2\int_{\Omega} u_1^2u_2^2dx,
\end{gather*}
and  assumption (A4) is satisfied with $\delta=1/2$. To
apply Theorem \ref{thm3.4}, we need to check that the initial data set that
satisfies conditions $E(0)>0$, $I(0)<0$ and
\begin{equation}
\Vert\phi_1\Vert_2^2+\Vert\phi_2\Vert_2^2>4E(0), \label{e3.27}
\end{equation}
by \eqref{e3.5} is not empty. Setting
\begin{equation}
\begin{gathered}
\alpha=\| \phi_1\|_2^2+\| \phi_2\|_2^2,\quad
\beta=\| \nabla\phi_1\|_2 ^2+\| \nabla\phi_2\|_2^2,\\
\gamma=\| \phi_1\phi_2\|_2^2,\quad 
\lambda =\| \varphi_1\|_2^2+\| \varphi_2\|_2^2.
\end{gathered}\label{e3.28}
\end{equation}
Then the above conditions $E(0)>0$, $I(0)<0$ and \eqref{e3.27} read as
follows
\begin{gather}
E(0)    =\frac{1}{2}(\alpha+\beta+\lambda)  -\frac{1}{2}\gamma>0,\label{e3.29}\\
I(0)    =\alpha+\beta-2\gamma<0,\label{e3.30}\\
\alpha   >2(\alpha+\beta+\lambda)  -2\gamma. \label{e3.31}
\end{gather}
Having \eqref{e3.30} in mind, we choose $\phi_1$ and $\phi_2$ such that
\begin{equation}
\alpha+\beta=2\gamma-\varepsilon\gamma, \label{e3.32}
\end{equation}
with $0<\varepsilon<2$. Thus \eqref{e3.30} is satisfied. 
At this moment, we consider two cases: (i) $0<\varepsilon\leq1$ 
and (ii) $1<\varepsilon<2$.

Case (i) $0<\varepsilon\leq1$. In this case, we further require $\phi_1$ and
$\phi_2$ to satisfy
$\alpha>-2(\varepsilon-1)  \gamma$,
and then, select $\lambda$ such that
\begin{equation}
0<\lambda<\frac{\alpha}{2}+(\varepsilon-1)  \gamma. \label{e3.33}
\end{equation}
Substituting \eqref{e3.32} into \eqref{e3.29} and $0<\varepsilon\leq1$, we see that
\[
2E(0)   =\alpha+\beta+\lambda-\gamma
  =\lambda-(\varepsilon-1)  \gamma   >0,
\]
this implies that \eqref{e3.29} is achieved. 
Since $\lambda<\frac{\alpha} {2}+(\varepsilon-1)  \gamma$ by \eqref{e3.33}, 
we deduce that
\[
\alpha   >2\lambda-2(\varepsilon-1)  \gamma
 =2(\alpha+\beta+\lambda)  -2\gamma,
\]
where the last equality is derived due to \eqref{e3.32}. 
Thus \eqref{e3.31} is obtained.

Case (ii) $1<\varepsilon<2$. In this case, we select $\lambda$ such that
\begin{equation}
(\varepsilon-1)  \gamma<\lambda<\frac{\alpha}{2}+(
\varepsilon-1)  \gamma. \label{e3.34}
\end{equation}
Similarly as in part (i), we see that the conditions \eqref{e3.29}-\eqref{e3.31} 
are satisfied. Therefore, from above arguments, the set of all initial data which
satisfy the conditions $E(0)>0$, $I(0)<0$ and \eqref{e3.27} is not empty.

Furthermore, although 
$\Vert\phi_1\Vert_2^2+\Vert\phi_2\Vert_2 ^2>4E(0)$ 
gives an upper bound of the initial energy $E(0)$. 
$E(0)$ can be chosen to be arbitrary positive provided that 
$\alpha=\| \phi _1\|_2^2+\| \phi_2\|_2^2$ is large
enough and $\beta,\gamma$ can be also larger accordingly to make sure
\eqref{e3.29}-\eqref{e3.31} is still satisfied.

Next, we give an example to illustrate the above discussion is workable.
Consider the problem \eqref{e3.26} with $\Omega=(0,4)$,
\[
\phi_1(x)=\begin{cases}
x,& 0<x<1,\\
x^2, & 1\leq x<3,\\
-9x+36,& 3\leq x<4;
\end{cases}
\quad 
\phi_2(x)=\begin{cases}
3x,& 0<x<1,\\
3,& 1\leq x<3,\\
-3x+12,& 3\leq x<4.
\end{cases}
\]
Then, from \eqref{e3.28} and \eqref{e3.32}, we have the following data
\begin{gather*}
\alpha   =\| \phi_1\|_2^2+\| \phi _2\|_2^2=99.73,\quad 
\beta=\| \nabla\phi _1\|_2^2+\| \nabla\phi_2\|_2^2=134.67\\
\gamma   =\| \phi_1\phi_2\|_2^2=583.2,\quad \varepsilon=1.598.
\end{gather*}
Now, based on \eqref{e3.34}, choose $\lambda$ such that
\[
348.8=(\varepsilon-1)  \gamma<\lambda=\| \varphi
_1\|_2^2+\| \varphi_2\|_2^2
<\frac{\alpha}{2}+(\varepsilon-1)  \gamma=398.65.
\]
Then
\begin{gather*}
E(0)   =\frac{1}{2}(\alpha+\beta+\lambda)-\frac{1}{2}\gamma
 =\frac{1}{2}(\lambda-348.8)  >0, \\
I(0)=\alpha+\beta-2\gamma=-932<0, \\
2(\alpha+\beta+\lambda)  -2\gamma=2(\lambda-348.8)<\alpha.
\end{gather*}
Thus Theorem \ref{thm3.4} is applicable.
\end{example}

\begin{example} \label{examp3.6} \rm
 Consider the system \eqref{e1.1}-\eqref{e1.3} in $\mathbb{R}^3$ with
\[
f_1(u_1,u_2)=4\lambda(u_1+\alpha u_2)  ^{3}+2\beta u_1u_2^2,\quad 
f_2(u_1,u_2)=4\alpha\lambda(u_1+\alpha u_2)  ^{3}+2\beta u_1^2u_2.
\]
Assume that $\lambda>0$, $\beta>0$ and $\alpha$ is any real number. Now we
have
\[
F(u_1,u_2)=\lambda(u_1+\alpha u_2)  ^{4}+2\beta u_1^2u_2^2.
\]
We see that (A4) is satisfied if $0<\delta\leq 1/2$. Thus Theorem \ref{thm3.4}
is applicable.
\end{example}

\subsection*{Acknowledgements}
The author would like thank the anonymous referees for their
comments. He is also grateful to Professor Liu Wenjun for his comments on
Example \ref{examp3.5}.

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