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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 216, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/216\hfil Blow-up of solutions]
{Blow-up of solutions for a system of nonlinear parabolic equations}

\author[S.-T. Wu \hfil EJDE-2013/216\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 August 21, 2013. Published September 30, 2013.}
\subjclass[2000]{35K55, 35K60}
\keywords{Blow-up; lower bound of blow-up time; parabolic problem}

\begin{abstract}
 The initial boundary value problem for a system of parabolic equations in a
 bounded domain is considered. We prove that, under suitable conditions on
 the nonlinearity and certain initial data, the lower bound for the blow-up
 time is determined if blow-up does occur. In addition, a criterion for
 blow-up to occur and conditions which ensure that blow-up does not occur are
 established.
\end{abstract}

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


\section{Introduction}

We consider the initial boundary value problem for the following nonlinear
parabolic problems:
\begin{gather}
u_{t}-\operatorname{div}(\rho _1(|\nabla u| ^2) \nabla u)
=f_1(u,v)\quad \text{in }\Omega \times [0,\infty ),  \label{e1.1}
\\
v_{t}-\operatorname{div}((\rho _2(|\nabla v|^2) \nabla v)
=f_2(u,v)\quad\text{in }\Omega \times [0,\infty ),  \label{e1.2}
\\
u(x,0) =u_{0}(x) ,\quad v(x,0)=v_{0}(x) ,\quad x\in \Omega ,  \label{e1.3} \\
u(x,t) =v(x,t) =0,\quad x\in \partial \Omega ,\;t>0,  \label{e1.4}
\end{gather}
where $\Omega $ is a bounded domain in $\mathbb{R}^N$ $(N\geq 1) $ with a
 smooth boundary $\partial \Omega $, $\rho _i$, $i=1,2$, are positive
 $C^1$ functions and $f_i(\cdot ,\cdot):\mathbb{R}^2\to \mathbb{R}$, $i=1,2$,
 are given functions which will be specified
later. $u_{0}(x) $, $v_{0}(x) $ are nonzero and
nonnegative functions.

Questions related to the blow-up phenomena of the solutions for the
nonlinear parabolic equations and systems have attracted considerable
attention in recent years. A natural question concerning the blow-up
properties is about whether the solution blows up and, if so, at what time 
$t^{\ast }$ blow-up occurs. In this direction, there is a vast literature to
deal with the blow-up time when the solution does blow up at finite time
 $t^{\ast }$ \cite{b1,b2,c1,f1,g1,k1,l1,l2,p1,p3},  \cite[page 3]{s1}. Yet, 
this blow-up time can seldom
be determined explicitly. Indeed, the methods used in the study of blow-up
very often have yielded only upper bound for $t^{\ast }$. However, a lower
bound on blow-up time is more important in some applied problems because of
the explosive nature of the solution. To the authors knowledge, some of the
first work on lower bounds for $t^{\ast }$ was by Weissler \cite{w1,w2}.
Recently, a number of papers deriving lower bounds for $t^{\ast }$ in
various problems have appeared, beginning with the paper of Payne and
Schaefer \cite{p4}. Payne et al. \cite{p5} considered the  single equation
\begin{equation*}
u_{t}-\operatorname{div}\big(\rho (|\nabla u|^2) \nabla u\big) =f(u).
\end{equation*}
Under certain conditions on the nonlineartities, they obtained a lower bound
for blow-up time if blow-up does occur. Additionally, a criterion for
blow-up and conditions which ensure that blow-up does not occur are obtained.

Motivated by previous works, in this study, we establish the lower bound and
the upper bound for problem \eqref{e1.1}-\eqref{e1.4} when blow-up does occur.
 Besides, the nonblow-up properties for a class of problem 
\eqref{e1.1}-\eqref{e1.4} are also
investigated. Our proof technique closely follows the arguments of 
\cite{p5},
with some modifications being needed for our problems. The paper is
organized as follows. In section 2, under suitable conditions on $\rho _i$,
$f_i$, $i=1,2$, the lower bound for the blow-up time is established if
blow-up occurs when $\Omega $ is a bounded domain in $\mathbb{R}^3$.
In Section 3, the nonblow-up phenomena are investigated. Finally, the
sufficient condition which guarantees the blow-up occurs is obtained and an
upper bound for the blow-up time is also given.

\section{Lower bound for the blow-up time}

In this section, we focus our attention to the lower bound time $t^{\ast }$
for the blow-up time of the solutions to problem \eqref{e1.1}-\eqref{e1.4}.
 For this purpose, we give the assumptions on $\rho _i$ and $f_i$, $i=1,2$ as
follows.
\begin{itemize}
\item[(A1)] $\rho _i(s)$, $i=1,2$ are nonnegative $C^1$ function for 
$s>0$ satisfying
\begin{equation*}
\rho _1(s)\geq b_1+b_2s^{q_1},\quad
\rho _2(s)\geq b_3+b_3s^{q_2},\quad q_1,\;q_2, \;b_i>0,\quad i=1-4.
\end{equation*}

\item[(A2)] Concerning the functions $f_1(u,v)$ and $f_2(u,v)$, we
take (see \cite{l3})
\begin{gather}
f_1(u,v) =\Big(a|u+v| ^{m-1}(u+v)+b|
u| ^{\frac{m-3}{2}}|v| ^{\frac{m+1}{2}}u\Big) ,  \label{e2.1} \\
f_2(u,v) =\Big(a|u+v| ^{m-1}(u+v)+b|
v| ^{\frac{m-3}{2}}|u| ^{\frac{m+1}{2}}v\Big) ,  \label{e2.2}
\end{gather}
where $a$, $b>0$ are constants and $m$ satisfies
\begin{equation*}
m>1,\text{ if }N=1,2\quad \text{or}\quad 
1<m\leq \frac{N+2}{N-2},\text{ if }N\geq 3.
\end{equation*}
\end{itemize}
One can easily verify that
\begin{equation*}
uf_1(u,v)+vf_2(u,v)=(m+1)F(u,v),\quad \forall (u,v)\in \mathbb{R}^2,
\end{equation*}
where
\begin{equation*}
F(u,v)=\frac{1}{m+1}\Big(a|u+v| ^{m+1}+2b|
uv| ^{\frac{m+1}{2}}\Big) .
\end{equation*}
As in \cite{l3}, we still have the following result.

\begin{lemma} \label{lem2.1}
There exists a positive constant $\beta $
 such that, for $\mathit{p>0}$,
\begin{equation*}
u^{p}f_1(u,v)+v^{p}f_2(u,v)\leq \beta (|u|
^{p+m}+|v| ^{p+m}) ,\quad \forall (u,v)\in \mathbb{R}^2.
\end{equation*}
We define
\begin{equation}
\begin{split}
\phi (t) &=\int_{\Omega }u^{2(n-1)(q_1+1)+2}dx+\int_{\Omega
}v^{2(n-1)(q_2+1)+2}dx   \\
&=\int_{\Omega }u^{\sigma _1}dx+\int_{\Omega }v^{\sigma _2}dx,
\end{split}\label{e2.3}
\end{equation}
where $\sigma _1=2(n-1)(q_1+1)+2$, $\sigma _2=2(n-1)(q_2+1)+2$ and
$n $ is a positive constant satisfying
\begin{equation}
\begin{split}
n >\max \Big\{& \frac{3(m-1)-2q_1}{2(q_1+1)},\frac{3(m-1)-2q_2}{
2(q_2+1)},\frac{3(m-1)-2(3q_1-2q_2) }{2(3q_1-2q_2+1)}, \\
&\frac{3(m-1)-2(3q_2-2q_1) }{2(3q_2-2q_1+1)} \Big\} .
\end{split} \label{e2.4}
\end{equation}
\end{lemma}

\begin{theorem} \label{thm2.2}
Suppose that {\rm (A1), (A2)}, \eqref{e2.4} hold and 
$\Omega \subset\mathbb{R}^3$ is a bounded domain. Assume further that 
$m-1>2\max (q_1,q_2) >0$ and $q_1>\frac{2}{3}q_2>\frac{4}{9}q_1>0$.
 Let $(u,v)$ be the nonnegative solution of
problem \eqref{e1.1}-\eqref{e1.4}, which become unbounded in the measure 
$\phi $ at time $t^{\ast }$, then $t^{\ast }$ is bounded below as
\begin{equation*}
t^{\ast }\geq \int_{\phi (0)}^{\infty }\frac{1}{\sum
_{i=1}^{4}k_i\phi (s)^{\mu _i}}ds,
\end{equation*}
where $k_i>0$  and $\mu _i>0$, $i=1-4$ are constnats given in the proof.
\end{theorem}

\begin{proof} Differentiating \eqref{e2.3} and using \eqref{e1.1}-\eqref{e1.2}, 
(A1) and Lemma \ref{lem2.1}, we obtain
\begin{equation}
\begin{split}
\phi '(t)
&= \sigma _1\int_{\Omega }u^{\sigma
_1-1}u_{t}dx+\sigma _2\int_{\Omega }v^{\sigma _2-1}v_{t}dx   \\
&= -\sigma _1(\sigma _1-1) \int_{\Omega }u^{\sigma
_1-2}\rho _1(|\nabla u| ^2) |
\nabla u| ^2dx+\sigma _1\int_{\Omega }u^{\sigma
_1-1}f_1(u,v)dx   \\
&\quad-\sigma _2(\sigma _2-1) \int_{\Omega }v^{\sigma
_2-2}\rho _2(|\nabla v| ^2) |
\nabla v| ^2dx+\sigma _2\int_{\Omega }v^{\sigma_2-1}f_2(u,v)dx   \\
&\leq -\sigma _1(\sigma _1-1) \int_{\Omega }u^{\sigma
_1-2}|\nabla u| ^2(b_1+b_2|
\nabla u| ^{2q_1}) dx\\
&\quad +\beta \sigma _1\int_{\Omega }(u^{m+\sigma _1-1}+v^{m+\sigma _1-1}) dx   \\
&\quad-\sigma _2(\sigma _2-1) \int_{\Omega }v^{\sigma _2-2}
 |\nabla v| ^2(b_3+b_4| \nabla v| ^{2q_2}) dx   \\
&\quad +\beta \sigma _2\int_{\Omega }(u^{m+\sigma _2-1}+v^{m+\sigma
_2-1}) dx.
\end{split} \label{e2.5}
\end{equation}
Dropping the terms $\sigma _1(\sigma _1-1) b_1\int_{\Omega
}u^{\sigma _1-2}|\nabla u| ^2dx$ and $\sigma
_2(\sigma _2-1) b_3\int_{\Omega }v^{\sigma
_2-2}|\nabla v| ^2dx$ on the right-hand side of
\eqref{e2.5} and using $|\nabla w^n| ^2=n^2w^{2(
n-1) }|\nabla w| ^2$, we deduce that
\begin{align*}
\phi '(t)
&\leq -\frac{\sigma _1(\sigma _1-1) b_2 }{n^{2(q_1+1) }}
 \int_{\Omega }|\nabla u^n| ^{2(q_1+1) }dx
+\beta \sigma _1\int_{\Omega }(u^{m+\sigma _1-1}+v^{m+\sigma _1-1}) dx \\
&\quad -\frac{\sigma _2(\sigma _2-1) b_4}{n^{2(
q_2+1) }}\int_{\Omega }|\nabla v^n| ^{2(
q_2+1) }dx+\beta \sigma _2\int_{\Omega }(u^{m+\sigma
_2-1}+v^{m+\sigma _2-1}) dx.
\end{align*}
For simplicity, setting $w_1=u^n$, $w_2=v^n$ and
$\gamma_i=m-1-2q_i>0$, $i=1,2$, then we obtain
\begin{equation}
\begin{split}
\phi '(t) 
&\leq -\frac{\sigma _1(\sigma _1-1) b_2
}{n^{2(q_1+1) }}\int_{\Omega }|\nabla w_1| ^{2(q_1+1) }dx\\
&\quad +\beta \sigma _1\int_{\Omega
}(w_1^{2(q_1+1) +\frac{\gamma _1}{n}}+w_2^{2(
q_1+1) +\frac{\gamma _1}{n}}) dx   \\
&\quad -\frac{\sigma _2(\sigma _2-1) b_4}{n^{2(
q_2+1) }}\int_{\Omega }|\nabla w_2| ^{2(
q_2+1) }dx+\beta \sigma _2\int_{\Omega }w_1^{2(
q_2+1) +\frac{\gamma _2}{n}}dx   \\
&\quad +\beta \sigma _2\int_{\Omega }w_2^{2(q_2+1) +\frac{
\gamma _2}{n}}dx.
\end{split}  \label{e2.6}
\end{equation}
Next, we will estimate the right-hand side of \eqref{e2.6}.
It follows from \cite[(2.12)]{p5} that
\begin{equation}
\int_{\Omega }w_1^{2(q_1+1) +\frac{\gamma _1}{n}}dx
\leq K_1\Big(\int_{\Omega }|\nabla w_1| ^{2(
q_1+1) }dx\Big) ^{2/3}
\Big(\int_{\Omega }w_1^{q_1+1+
\frac{3\gamma _1}{2n}}dx\Big) ^{2/3},  \label{e2.7}
\end{equation}
where $K_1=\alpha \lambda _1^{-\frac{4q_1+1}{6}}(q_1+1)
^{\frac{4(q_1+1) }{3}}$,
 $\alpha =4^{1/3}\cdot 3^{-1/2}\cdot \pi ^{-2/3}$ and $\lambda _1$ is the first
eigenvalue in the fixed membrane problem
\begin{equation*}
\Delta w+\lambda w=0,\quad w>0\text{ in }\Omega,\quad \text{and}\quad
w=0\text{ on } \partial \Omega .
\end{equation*}
By using H\"{o}lder inequality and \eqref{e2.3}, we obtain
\begin{equation}
\begin{split}
\int_{\Omega }w_1^{q_1+1+\frac{3\gamma _1}{2n}}dx
&= \int_{\Omega }u^{n(q_1+1) +\frac{3\gamma _1}{2}}dx\\
&\leq \Big(\int_{\Omega }u^{\sigma _1}dx\Big) ^{\mu _1}\cdot |\Omega
| ^{1-\mu _1}\\
&\leq \phi (t)^{\mu _1}\cdot |\Omega | ^{1-\mu _1},
\end{split} \label{e2.8}
\end{equation}
which together with \eqref{e2.7} implies
\[
\int_{\Omega }w_1^{2(q_1+1) +\frac{\gamma _1}{n}}dx
\leq K_1|\Omega | ^{\frac{2(1-\mu _1) }{3}
}\phi (t)^{\frac{2\mu _1}{3}}(\int_{\Omega }|\nabla
w_1| ^{2(q_1+1) }dx) ^{2/3}.
\]
with $\mu _1=\frac{2n(q_1+1) +3\gamma _1}{2\sigma _1}$,
we note that $\mu _1<1$ in view of \eqref{e2.4}.
Further, thanks to the inequality
\begin{equation}
x^{r}y^{s}\leq rx+sy,\quad r+s=1,\quad x,y\geq 0,  \label{e2.9}
\end{equation}
we obtain, for $\alpha _1>0$,
\begin{equation}
\int_{\Omega }w_1^{2(q_1+1) +\frac{\gamma _1}{n}}dx\leq
K_1|\Omega | ^{\frac{2(1-\mu _1) }{3}}
\Big[ \frac{1}{3\alpha _1^2}\phi (t)^{2\mu _1}+\frac{2\alpha _1}{3}
\int_{\Omega }|\nabla w_1| ^{2(q_1+1) }dx
\Big] .  \label{e2.10}
\end{equation}
and similarly
\begin{equation}
\int_{\Omega }w_2^{2(q_2+1) +\frac{\gamma _2}{n}}dx\leq
K_2|\Omega | ^{\frac{2(1-\mu _2) }{3}}
\Big[ \frac{1}{3\alpha _2^2}\phi (t)^{2\mu _2}+\frac{2\alpha _2}{3}
\int_{\Omega }|\nabla w_2| ^{2(q_2+1) }dx
\Big] ,  \label{e2.11}
\end{equation}
where $\alpha _2>0$, $K_2=\alpha \lambda _1^{-\frac{4q_2+1}{6}
}(q_2+1) ^{\frac{4(q_2+1) }{3}}$ and $\mu _2=
\frac{2n(q_2+1) +3\gamma _2}{2\sigma _2}<1$.

To estimate the other two terms in the right hand side of \eqref{e2.6}, we
use H\"{o}lder inequality and the following result 
(see \cite[(2.7)-(2.10)]{p5})
\begin{equation}
\int_{\Omega }w^{4(q+1)}dx\leq \alpha ^3(q+1) ^{4(q+1)}\lambda
_1^{-\frac{4q+1}{2}}\big(\int_{\Omega }|\nabla w|
^{2(q+1)}dx\Big) ^2,\quad q>0,  \label{e2.12}
\end{equation}
to obtain
\begin{equation}
\begin{split}
&\int_{\Omega }w_2^{2(q_1+1) +\frac{\gamma _1}{n}}dx
\\
&= \int_{\Omega }w_2^{\frac{4(q_2+1) }{3}}\cdot
w_2^{2(q_1+1) -\frac{4(q_2+1) }{3}+\frac{
\gamma _1}{n}}dx\\
&\leq \big(\int_{\Omega }w_2^{4(q_2+1) }dx\big) ^{1/3}
\Big(\int_{\Omega} w_2^{3 q_1-2q_2+1+\frac{3\gamma _1}{2n}} dx\big) ^{2/3}   \\
&\leq K_2\Big(\int_{\Omega }|\nabla w_2|
^{2(q_2+1) }dx\Big) ^{2/3}\Big(\int_{\Omega
}w_2^{3q_1-2q_2+1+\frac{3\gamma _1}{2n}}dx\Big) ^{2/3}.
\end{split}\label{e2.13}
\end{equation}
As in deriving \eqref{e2.8}, we see that
\begin{equation}
\begin{split}
\int_{\Omega }w_2^{3q_1-2q_2+1+\frac{3\gamma _1}{2n}}dx
&= \int_{\Omega }v^{n(3q_1-2q_2+1) +\frac{3\gamma _1}{2}}dx\\
&\leq (\int_{\Omega }v^{\sigma _2}dx) ^{\mu _3}\cdot |\Omega | ^{1-\mu _3}   \\
&\leq \phi (t)^{\mu _3}\cdot |\Omega | ^{1-\mu _3}
\end{split}\label{e2.14}
\end{equation}
where $\mu _3=\frac{2n(3q_1-2q_2+1) +3\gamma _1}{2\sigma
_2}<1$. Substituting \eqref{e2.14} into \eqref{e2.13} and using \eqref{e2.9}
once more, we obtain, for $\alpha _3>0$,
\begin{equation}
\int_{\Omega }w_2^{2(q_1+1) +\frac{\gamma _1}{n}}dx\leq
K_2|\Omega | ^{\frac{2(1-\mu _3) }{3}}
\Big[ \frac{1}{3\alpha _3^2}\phi (t)^{2\mu _3}+\frac{2\alpha _3}{3}
\int_{\Omega }|\nabla w_2| ^{2(q_2+1) }dx
\Big] .  \label{e2.15}
\end{equation}
and similarly
\begin{equation}
\int_{\Omega }w_1^{2(q_2+1) +\frac{\gamma _2}{n}}dx\leq
K_1|\Omega | ^{\frac{2(1-\mu _4) }{3}}
\Big[ \frac{1}{3\alpha _4^2}\phi (t)^{2\mu _4}+\frac{2\alpha _4}{3}
\int_{\Omega }|\nabla w_1| ^{2(q_1+1) }dx
\Big] ,  \label{e2.16}
\end{equation}
where $\alpha _4>0$ and $\mu _4=\frac{2n(3q_2-2q_1+1)
+3\gamma _2}{2\sigma _1}<1$.
Combining \eqref{e2.10}, \eqref{e2.11}, \eqref{e2.15} and \eqref{e2.16}
with \eqref{e2.6}, we conclude that
\begin{align*}
\phi '(t)
&\leq -C_1\int_{\Omega }|\nabla
w_1| ^{2(q_1+1) }dx-C_2\int_{\Omega }|\nabla w_2| ^{2(q_2+1) }dx\\
&\quad +k_1\phi (t)^{2\mu_1}  +k_2\phi (t)^{2\mu _2}+k_3\phi (t)^{2\mu _3}
 +k_4\phi (t)^{2\mu_4},
\end{align*}
where
\begin{gather*}
C_1 = \frac{\sigma _1(\sigma _1-1) b_2}{n^{2(
q_1+1) }}-\frac{2\alpha _1K_1\beta \sigma _1}{3}|
\Omega | ^{\frac{2(1-\mu _1) }{3}}-\frac{2\alpha
_4K_1\beta \sigma _2}{3}|\Omega | ^{\frac{2(
1-\mu _4) }{3}}, \\
C_2 = \frac{\sigma _2(\sigma _2-1) b_4}{n^{2(
q_2+1) }}-\frac{2\alpha _2K_2\beta \sigma _1}{3}|
\Omega | ^{\frac{2(1-\mu _2) }{3}}-\frac{2\alpha
_3K_2\beta \sigma _2}{3}|\Omega | ^{\frac{2(
1-\mu _3) }{3}}, \\
k_1 = \frac{K_1|\Omega | ^{\frac{2(1-\mu_1) }{3}}\beta \sigma _1}{3\alpha _1^2},\quad
k_2=\frac{K_2|\Omega | ^{\frac{2(1-\mu _2) }{3} }\beta \sigma _2}{3\alpha _2^2},\\
k_3=\frac{K_2|\Omega| ^{\frac{2(1-\mu _3) }{3}}\beta \sigma _1}{3\alpha _3^2}, \quad
k_4 = \frac{K_1|\Omega | ^{\frac{2(1-\mu_4) }{3}}\beta \sigma _2}{3\alpha _4^2}.
\end{gather*}
Now, setting
$\alpha _1=\alpha _2$, $\alpha _3=\alpha _4$,
and choosing $\alpha _1,\alpha _3$ such that $C_1=0$ and $C_2=0$,
hence, we have
\begin{equation}
\phi '(t)\leq g(\phi ),  \label{e2.17}
\end{equation}
where
\begin{equation*}
g(s)=k_1s^{2\mu _1}+k_2s^{2\mu _2}+k_3s^{2\mu _3}+k_4s^{2\mu
_4}.
\end{equation*}
An integration of \eqref{e2.17} from $0$ to $t$ leads to
\begin{equation*}
\int_{\phi (0)}^{\phi (t)}\frac{ds}{g(s)}\leq t,
\end{equation*}
so that if $(u,v)$ blows up in the measure of $\phi $ as $t\to t^{\ast }$,
we derive the lower bound
\begin{equation*}
\int_{\phi (0)}^{\infty }\frac{ds}{g(s)}\leq t^{\ast },
\end{equation*}
and Theorem \ref{thm2.2} is proved.
Clearly, the integral is bounded since $2\mu _1>1$.
\end{proof}

\section{Non blow-up case}

In this section, we consider the non blow-up property of problem 
\eqref{e1.1}-\eqref{e1.4} when $2\max (q_1,q_2) >m-1>0$. To achieve this, 
we define the auxiliary function
\begin{equation}
\phi (t)=\frac{1}{2}\int_{\Omega }u^2dx+\frac{1}{2}\int_{\Omega }v^2dx.
\label{e3.1}
\end{equation}

\begin{theorem} \label{thm3.1} 
Suppose that {\rm (A1), (A2)} hold and that 
$2\max (q_1,q_2) >m-1>0$.
 Let $(u,v)$ be the nonnegative solution of problem \eqref{e1.1}-\eqref{e1.4}, 
then $(u,v)$ can not blow up in the measure $\phi $ in finite time.
\end{theorem}

\begin{proof} 
From\eqref{e3.1}, \eqref{e1.1}, \eqref{e1.2} and (A2), we have
\begin{equation}
\begin{split}
\phi '(t)
&= \int_{\Omega }uu_{t}dx+\int_{\Omega }vv_{t}dx \\
&\leq -\int_{\Omega }|\nabla u| ^2(b_1+b_2|\nabla u| ^{2q_1}) dx
-\int_{\Omega }|\nabla v| ^2(b_3+b_4|\nabla v| ^{2q_2}) dx   \\
&\quad +\beta \int_{\Omega }(u^{m+1}+v^{m+1}) dx   \\
&\leq \int_{\Omega }(\beta u^{m+1}-b_2|\nabla u| ^{2(q_1+1) }) dx
+\int_{\Omega }(\beta v^{m+1}-b_4|\nabla v| ^{2(q_2+1)}) dx   \\
&\leq \int_{\Omega }\Big(\beta u^{m+1}-b_2(\frac{\lambda _1}{
(q_1+1) ^2}) ^{q_1+1}u^{2(q_1+1) }\Big) dx   \\
&\quad +\int_{\Omega }\big(\beta v^{m+1}-b_4(\frac{\lambda _1}{(
q_2+1) ^2}) ^{q_2+1}v^{2(q_2+1) }\big) dx,
\end{split} \label{e3.2}
\end{equation}
where the last inequality is obtained by using \cite[(2.10]{p5}.
For $q>0$,
\begin{equation*}
\int_{\Omega }w^{2(q+1) }dx\leq (\frac{(q+1)^2}{\lambda _1}) ^{q+1}
\int_{\Omega }|\nabla w| ^{2(q+1) }dx,
\end{equation*}
where $\lambda _1$ is the first eigenvalue in the fixed membrane problem,
as defined in Section 2. Employing H\"{o}lder inequality, we have
\begin{gather}
\int_{\Omega }u^{m+1}dx
\leq \Big(\int_{\Omega }u^{2(
q_1+1) }dx\Big) ^{\frac{m+1}{2(q_1+1)}}\cdot |\Omega
| ^{\frac{2q_1-m+1}{2(q_1+1)}},  \label{e3.3}
\\
\int_{\Omega }v^{m+1}dx \leq \big(\int_{\Omega }v^{2(
q_2+1) }dx\Big) ^{\frac{m+1}{2(q_{21}+1)}}\cdot |\Omega
| ^{\frac{2q_2-m+1}{2(q_2+1)}},  \label{e3.4}
\\
\int_{\Omega }u^2dx\leq \Big(\int_{\Omega }u^{m+1}dx\Big) ^{\frac{2}{m+1}}
\cdot |\Omega | ^{\frac{m-1}{m+1}}.  \label{e3.5}
\end{gather}
Inserting \eqref{e3.3}-\eqref{e3.5} into \eqref{e3.2}, we see that
\begin{equation}
\begin{split}
\phi '(t)
&\leq \int_{\Omega }u^{m+1}dx(\beta -M_1(
\int_{\Omega }u^2dx) ^{\frac{2q_1-m+1}{2}}) dx   \\
&\quad +2\int_{\Omega }v^{m+1}dx(\beta -M_2(\int_{\Omega
}v^2dx) ^{\frac{2q_2-m+1}{2}}) dx
\end{split}  \label{e3.6}
\end{equation}
where
\[
M_1=b_2(\frac{\lambda _1}{(q_1+1) ^2}) ^{q_1+1}|\Omega |
 ^{-\frac{2q_1-m+1}{2}},\quad
M_2=b_4(\frac{\lambda _1}{(q_2+1) ^2}) ^{q_2+1}|\Omega | ^{-\frac{2q_2-m+1}{2}}.
\]
Apparently, if $(u,v)$ blows up in the $\phi $ measure at some time $t$ then
$\phi '(t)$ would be negative which leads to a contradiction. Thus,
the solution $(u,v)$ can not blow up in the measure $\phi $. The proof is
complete.
\end{proof}

\section{Criterion for blow-up}

In this section, we investigate the blow up properties of solutions for 
 \eqref{e1.1}-\eqref{e1.4} with
\begin{equation} \label{e4.1}
\rho _1(s)=b_1+b_2s^{q_1},\quad
\rho _2(s)=b_3+b_3s^{q_2},\quad
q_1,q_2,b_i>0,\quad i=1-4.
\end{equation}
For this purpose, we first define
\begin{equation}
\phi (t)=\frac{1}{2}\int_{\Omega }u^2dx+\frac{1}{2}\int_{\Omega }v^2dx
\label{e4.2}
\end{equation}
and
\begin{equation}
\begin{split}
\psi (t) &= -\frac{b_1}{2}\| \nabla u\| _2^2-\frac{
b_2}{2(q_1+1)}\int_{\Omega }|\nabla u|
^{2(q_1+1)}dx-\frac{b_3}{2}\| \nabla v\| _2^2
 \\
&\quad -\frac{b_4}{2(q_2+1)}\int_{\Omega }|\nabla v|
^{2(q_2+1)}dx+\int_{\Omega }F(u,v)dx,
\end{split} \label{e4.3}
\end{equation}
where $\| \cdot \| _2$ is the $L^2(\Omega )$-norm.

\begin{theorem} \label{thm4.1} 
Suppose that \eqref{e4.1} and {\rm (A2)} hold. Assume
further that $m-1>2\max (q_1,q_2)\geq 0$ and $\psi (0)>0$.
 If (u,v) is the non-negative solution of problem \eqref{e1.1}-\eqref{e1.4}, then
the solution blows up at finite time $t^{\ast }$ with 
\begin{equation*}
t^{\ast }\leq \frac{\phi (0)^{-2m-1}}{(2m+1)(m+1)}.
\end{equation*}
\end{theorem}

\begin{proof} From \eqref{e4.1}-\eqref{e4.3}, we have
\begin{equation}
\begin{split}
\phi '(t)
&= -\int_{\Omega }|\nabla u|^2(b_1+b_2|\nabla u| ^{2q_1}) dx
-\int_{\Omega }|\nabla v| ^2(b_3+b_4|\nabla v| ^{2q_2}) dx   \\
&\quad +(m+1)\int_{\Omega }F(u,v)dx   \\
&\geq (m+1)\Big[ -\frac{b_1}{2}\int_{\Omega }|\nabla
u| ^2dx-\frac{b_2}{2(q_1+1)}\int_{\Omega }|\nabla u| ^{2(q_1+1)}dx\\
&\quad -\frac{b_3}{2}\| \nabla v\|_2^2
    -\frac{b_4}{2(q_2+1)}\int_{\Omega }|\nabla v| ^{2(q_2+1)}dx
 +\int_{\Omega }F(u,v)dx\Big]    \\
&= (m+1)\psi (t),
\end{split}  \label{e4.4}
\end{equation}
and
\begin{equation}
\begin{split}
\psi '(t)
&= -b_1\int_{\Omega }\nabla u\cdot \nabla
u_{t}dx-b_2\int_{\Omega }|\nabla u| ^{2q_1}\nabla
u\cdot \nabla u_{t}dx-b_3\int_{\Omega }\nabla v\cdot \nabla v_{t}dx
\\
&\quad -b_4\int_{\Omega }|\nabla v| ^{2q_2}\nabla v\cdot
\nabla v_{t}dx-\int_{\Omega }a|u+v| ^{m-1}(u+v)(
u_{t}+v_{t}) dx   \\
&\quad -b\int_{\Omega }\big(|u| ^{\frac{m-3}{2}}|
v| ^{\frac{m+1}{2}}uu_{t}+|v| ^{\frac{m-3}{2}
}|u| ^{\frac{m+1}{2}}vv_{t}\big) dx   \\
&= \int_{\Omega }(u_{t}^2+v_{t}^2) dx\geq 0. 
\end{split} \label{e4.5}
\end{equation}
This, together with $\psi (0)>0$, implies that $\psi (t)\geq \psi (0)>0$,
for $t\geq 0$. By using H\"{o}lder inequality, Schwarz inequality,
 \eqref{e4.2} and \eqref{e4.5}, we obtain
\begin{equation}
\begin{split}
(\phi '(t)) ^2
&= \Big(\int_{\Omega }uu_{t}dx+\int_{\Omega }vv_{t}dx\Big) ^2   \\
&\leq \| u\| _2^2\| u_{t}\|
_2^2+\| v\| _2^2\| v_{t}\|
_2^2+\| u\| _2^2\| v_{t}\|
_2^2+\| u\| _2^2\| u_{t}\| _2^2
 \\
&= \frac{1}{2}\phi (t)\psi '(t).
\end{split}  \label{e4.6}
\end{equation}
Then, using \eqref{e4.4} and \eqref{e4.6}, we deduce that
\begin{equation*}
\phi '(t)\psi (t)\leq \frac{1}{m+1}(\phi '(t))
^2\leq \frac{1}{2(m+1)}\phi \psi '(t),
\end{equation*}
which implies that
\begin{equation}
(\psi (t)\phi (t)^{-2m-2}) '\geq 0.  \label{e4.7}
\end{equation}
An integration of \eqref{e4.7} from $0$ to $t$ gives to
\begin{equation}
\psi (t)\phi (t)^{-2m-2}\geq \psi (0)\phi (0)^{-2m-2}\equiv M.  \label{e4.8}
\end{equation}
Combining \eqref{e4.4} with \eqref{e4.8} and integrating the resultant
differential inequality, we have
\begin{equation}
\phi (t)^{-2m-1}\leq \phi (0)^{-2m-1}-(2m+1)(m+1)Mt  \label{e4.9}
\end{equation}
Since $\phi (0) >0$, \eqref{e4.9} shows that $\phi $
becomes infinite in a finite time
\begin{equation*}
t^{\ast }\leq T=\frac{\phi (0)^{-2m-1}}{(2m+1)(m+1)}.
\end{equation*}
This completes the proof.
\end{proof}

\subsection*{Acknowledgments} 
The authors would like to thank  the
anonymous referees for their valuable comments and useful suggestions on
this work.

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\end{document}