\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 10}$, \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 $$\begin{split} \phi (t) &=\int_{\Omega }u^{2(n-1)(q_1+1)+2}dx+\int_{\Omega }v^{2(n-1)(q_2+1)+2}dx \\ &=\int_{\Omega }u^{\sigma _1}dx+\int_{\Omega }v^{\sigma _2}dx, \end{split}\label{e2.3}$$ 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 $$\begin{split} n >\max \Big\{& \frac{3(m-1)-2q_1}{2(q_1+1)},\frac{3(m-1)-2q_2}{ 2(q_2+1)},\frac{3(m-1)-2(3q_1-2q_2) }{2(3q_1-2q_2+1)}, \\ &\frac{3(m-1)-2(3q_2-2q_1) }{2(3q_2-2q_1+1)} \Big\} . \end{split} \label{e2.4}$$ \end{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 $$\begin{split} \phi '(t) &= \sigma _1\int_{\Omega }u^{\sigma _1-1}u_{t}dx+\sigma _2\int_{\Omega }v^{\sigma _2-1}v_{t}dx \\ &= -\sigma _1(\sigma _1-1) \int_{\Omega }u^{\sigma _1-2}\rho _1(|\nabla u| ^2) | \nabla u| ^2dx+\sigma _1\int_{\Omega }u^{\sigma _1-1}f_1(u,v)dx \\ &\quad-\sigma _2(\sigma _2-1) \int_{\Omega }v^{\sigma _2-2}\rho _2(|\nabla v| ^2) | \nabla v| ^2dx+\sigma _2\int_{\Omega }v^{\sigma_2-1}f_2(u,v)dx \\ &\leq -\sigma _1(\sigma _1-1) \int_{\Omega }u^{\sigma _1-2}|\nabla u| ^2(b_1+b_2| \nabla u| ^{2q_1}) dx\\ &\quad +\beta \sigma _1\int_{\Omega }(u^{m+\sigma _1-1}+v^{m+\sigma _1-1}) dx \\ &\quad-\sigma _2(\sigma _2-1) \int_{\Omega }v^{\sigma _2-2} |\nabla v| ^2(b_3+b_4| \nabla v| ^{2q_2}) dx \\ &\quad +\beta \sigma _2\int_{\Omega }(u^{m+\sigma _2-1}+v^{m+\sigma _2-1}) dx. \end{split} \label{e2.5}$$ 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$|\nabla w^n| ^2=n^2w^{2( n-1) }|\nabla w| ^2, we deduce that \begin{align*} \phi '(t) &\leq -\frac{\sigma _1(\sigma _1-1) b_2 }{n^{2(q_1+1) }} \int_{\Omega }|\nabla u^n| ^{2(q_1+1) }dx +\beta \sigma _1\int_{\Omega }(u^{m+\sigma _1-1}+v^{m+\sigma _1-1}) dx \\ &\quad -\frac{\sigma _2(\sigma _2-1) b_4}{n^{2( q_2+1) }}\int_{\Omega }|\nabla v^n| ^{2( q_2+1) }dx+\beta \sigma _2\int_{\Omega }(u^{m+\sigma _2-1}+v^{m+\sigma _2-1}) dx. \end{align*} For simplicity, settingw_1=u^n$,$w_2=v^n$and$\gamma_i=m-1-2q_i>0$,$i=1,2$, then we obtain $$\begin{split} \phi '(t) &\leq -\frac{\sigma _1(\sigma _1-1) b_2 }{n^{2(q_1+1) }}\int_{\Omega }|\nabla w_1| ^{2(q_1+1) }dx\\ &\quad +\beta \sigma _1\int_{\Omega }(w_1^{2(q_1+1) +\frac{\gamma _1}{n}}+w_2^{2( q_1+1) +\frac{\gamma _1}{n}}) dx \\ &\quad -\frac{\sigma _2(\sigma _2-1) b_4}{n^{2( q_2+1) }}\int_{\Omega }|\nabla w_2| ^{2( q_2+1) }dx+\beta \sigma _2\int_{\Omega }w_1^{2( q_2+1) +\frac{\gamma _2}{n}}dx \\ &\quad +\beta \sigma _2\int_{\Omega }w_2^{2(q_2+1) +\frac{ \gamma _2}{n}}dx. \end{split} \label{e2.6}$$ Next, we will estimate the right-hand side of \eqref{e2.6}. It follows from \cite[(2.12)]{p5} that $$\int_{\Omega }w_1^{2(q_1+1) +\frac{\gamma _1}{n}}dx \leq K_1\Big(\int_{\Omega }|\nabla w_1| ^{2( q_1+1) }dx\Big) ^{2/3} \Big(\int_{\Omega }w_1^{q_1+1+ \frac{3\gamma _1}{2n}}dx\Big) ^{2/3}, \label{e2.7}$$ 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 $$\begin{split} \int_{\Omega }w_1^{q_1+1+\frac{3\gamma _1}{2n}}dx &= \int_{\Omega }u^{n(q_1+1) +\frac{3\gamma _1}{2}}dx\\ &\leq \Big(\int_{\Omega }u^{\sigma _1}dx\Big) ^{\mu _1}\cdot |\Omega | ^{1-\mu _1}\\ &\leq \phi (t)^{\mu _1}\cdot |\Omega | ^{1-\mu _1}, \end{split} \label{e2.8}$$ 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 $$x^{r}y^{s}\leq rx+sy,\quad r+s=1,\quad x,y\geq 0, \label{e2.9}$$ we obtain, for$\alpha _1>0$, $$\int_{\Omega }w_1^{2(q_1+1) +\frac{\gamma _1}{n}}dx\leq K_1|\Omega | ^{\frac{2(1-\mu _1) }{3}} \Big[ \frac{1}{3\alpha _1^2}\phi (t)^{2\mu _1}+\frac{2\alpha _1}{3} \int_{\Omega }|\nabla w_1| ^{2(q_1+1) }dx \Big] . \label{e2.10}$$ and similarly $$\int_{\Omega }w_2^{2(q_2+1) +\frac{\gamma _2}{n}}dx\leq K_2|\Omega | ^{\frac{2(1-\mu _2) }{3}} \Big[ \frac{1}{3\alpha _2^2}\phi (t)^{2\mu _2}+\frac{2\alpha _2}{3} \int_{\Omega }|\nabla w_2| ^{2(q_2+1) }dx \Big] , \label{e2.11}$$ 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}) $$\int_{\Omega }w^{4(q+1)}dx\leq \alpha ^3(q+1) ^{4(q+1)}\lambda _1^{-\frac{4q+1}{2}}\big(\int_{\Omega }|\nabla w| ^{2(q+1)}dx\Big) ^2,\quad q>0, \label{e2.12}$$ to obtain $$\begin{split} &\int_{\Omega }w_2^{2(q_1+1) +\frac{\gamma _1}{n}}dx \\ &= \int_{\Omega }w_2^{\frac{4(q_2+1) }{3}}\cdot w_2^{2(q_1+1) -\frac{4(q_2+1) }{3}+\frac{ \gamma _1}{n}}dx\\ &\leq \big(\int_{\Omega }w_2^{4(q_2+1) }dx\big) ^{1/3} \Big(\int_{\Omega} w_2^{3 q_1-2q_2+1+\frac{3\gamma _1}{2n}} dx\big) ^{2/3} \\ &\leq K_2\Big(\int_{\Omega }|\nabla w_2| ^{2(q_2+1) }dx\Big) ^{2/3}\Big(\int_{\Omega }w_2^{3q_1-2q_2+1+\frac{3\gamma _1}{2n}}dx\Big) ^{2/3}. \end{split}\label{e2.13}$$ As in deriving \eqref{e2.8}, we see that $$\begin{split} \int_{\Omega }w_2^{3q_1-2q_2+1+\frac{3\gamma _1}{2n}}dx &= \int_{\Omega }v^{n(3q_1-2q_2+1) +\frac{3\gamma _1}{2}}dx\\ &\leq (\int_{\Omega }v^{\sigma _2}dx) ^{\mu _3}\cdot |\Omega | ^{1-\mu _3} \\ &\leq \phi (t)^{\mu _3}\cdot |\Omega | ^{1-\mu _3} \end{split}\label{e2.14}$$ 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$\alpha _3>0$, $$\int_{\Omega }w_2^{2(q_1+1) +\frac{\gamma _1}{n}}dx\leq K_2|\Omega | ^{\frac{2(1-\mu _3) }{3}} \Big[ \frac{1}{3\alpha _3^2}\phi (t)^{2\mu _3}+\frac{2\alpha _3}{3} \int_{\Omega }|\nabla w_2| ^{2(q_2+1) }dx \Big] . \label{e2.15}$$ and similarly $$\int_{\Omega }w_1^{2(q_2+1) +\frac{\gamma _2}{n}}dx\leq K_1|\Omega | ^{\frac{2(1-\mu _4) }{3}} \Big[ \frac{1}{3\alpha _4^2}\phi (t)^{2\mu _4}+\frac{2\alpha _4}{3} \int_{\Omega }|\nabla w_1| ^{2(q_1+1) }dx \Big] , \label{e2.16}$$ 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 $$\phi '(t)\leq g(\phi ), \label{e2.17}$$ 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 $$\phi (t)=\frac{1}{2}\int_{\Omega }u^2dx+\frac{1}{2}\int_{\Omega }v^2dx. \label{e3.1}$$ \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 $$\begin{split} \phi '(t) &= \int_{\Omega }uu_{t}dx+\int_{\Omega }vv_{t}dx \\ &\leq -\int_{\Omega }|\nabla u| ^2(b_1+b_2|\nabla u| ^{2q_1}) dx -\int_{\Omega }|\nabla v| ^2(b_3+b_4|\nabla v| ^{2q_2}) dx \\ &\quad +\beta \int_{\Omega }(u^{m+1}+v^{m+1}) dx \\ &\leq \int_{\Omega }(\beta u^{m+1}-b_2|\nabla u| ^{2(q_1+1) }) dx +\int_{\Omega }(\beta v^{m+1}-b_4|\nabla v| ^{2(q_2+1)}) dx \\ &\leq \int_{\Omega }\Big(\beta u^{m+1}-b_2(\frac{\lambda _1}{ (q_1+1) ^2}) ^{q_1+1}u^{2(q_1+1) }\Big) dx \\ &\quad +\int_{\Omega }\big(\beta v^{m+1}-b_4(\frac{\lambda _1}{( q_2+1) ^2}) ^{q_2+1}v^{2(q_2+1) }\big) dx, \end{split} \label{e3.2}$$ 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 $$\begin{split} \phi '(t) &\leq \int_{\Omega }u^{m+1}dx(\beta -M_1( \int_{\Omega }u^2dx) ^{\frac{2q_1-m+1}{2}}) dx \\ &\quad +2\int_{\Omega }v^{m+1}dx(\beta -M_2(\int_{\Omega }v^2dx) ^{\frac{2q_2-m+1}{2}}) dx \end{split} \label{e3.6}$$ 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 $$\label{e4.1} \rho _1(s)=b_1+b_2s^{q_1},\quad \rho _2(s)=b_3+b_3s^{q_2},\quad q_1,q_2,b_i>0,\quad i=1-4.$$ For this purpose, we first define $$\phi (t)=\frac{1}{2}\int_{\Omega }u^2dx+\frac{1}{2}\int_{\Omega }v^2dx \label{e4.2}$$ and $$\begin{split} \psi (t) &= -\frac{b_1}{2}\| \nabla u\| _2^2-\frac{ b_2}{2(q_1+1)}\int_{\Omega }|\nabla u| ^{2(q_1+1)}dx-\frac{b_3}{2}\| \nabla v\| _2^2 \\ &\quad -\frac{b_4}{2(q_2+1)}\int_{\Omega }|\nabla v| ^{2(q_2+1)}dx+\int_{\Omega }F(u,v)dx, \end{split} \label{e4.3}$$ 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 $$\begin{split} \phi '(t) &= -\int_{\Omega }|\nabla u|^2(b_1+b_2|\nabla u| ^{2q_1}) dx -\int_{\Omega }|\nabla v| ^2(b_3+b_4|\nabla v| ^{2q_2}) dx \\ &\quad +(m+1)\int_{\Omega }F(u,v)dx \\ &\geq (m+1)\Big[ -\frac{b_1}{2}\int_{\Omega }|\nabla u| ^2dx-\frac{b_2}{2(q_1+1)}\int_{\Omega }|\nabla u| ^{2(q_1+1)}dx\\ &\quad -\frac{b_3}{2}\| \nabla v\|_2^2 -\frac{b_4}{2(q_2+1)}\int_{\Omega }|\nabla v| ^{2(q_2+1)}dx +\int_{\Omega }F(u,v)dx\Big] \\ &= (m+1)\psi (t), \end{split} \label{e4.4}$$ and $$\begin{split} \psi '(t) &= -b_1\int_{\Omega }\nabla u\cdot \nabla u_{t}dx-b_2\int_{\Omega }|\nabla u| ^{2q_1}\nabla u\cdot \nabla u_{t}dx-b_3\int_{\Omega }\nabla v\cdot \nabla v_{t}dx \\ &\quad -b_4\int_{\Omega }|\nabla v| ^{2q_2}\nabla v\cdot \nabla v_{t}dx-\int_{\Omega }a|u+v| ^{m-1}(u+v)( u_{t}+v_{t}) dx \\ &\quad -b\int_{\Omega }\big(|u| ^{\frac{m-3}{2}}| v| ^{\frac{m+1}{2}}uu_{t}+|v| ^{\frac{m-3}{2} }|u| ^{\frac{m+1}{2}}vv_{t}\big) dx \\ &= \int_{\Omega }(u_{t}^2+v_{t}^2) dx\geq 0. \end{split} \label{e4.5}$$ 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 $$\begin{split} (\phi '(t)) ^2 &= \Big(\int_{\Omega }uu_{t}dx+\int_{\Omega }vv_{t}dx\Big) ^2 \\ &\leq \| u\| _2^2\| u_{t}\| _2^2+\| v\| _2^2\| v_{t}\| _2^2+\| u\| _2^2\| v_{t}\| _2^2+\| u\| _2^2\| u_{t}\| _2^2 \\ &= \frac{1}{2}\phi (t)\psi '(t). \end{split} \label{e4.6}$$ 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 $$(\psi (t)\phi (t)^{-2m-2}) '\geq 0. \label{e4.7}$$ An integration of \eqref{e4.7} from$0$to$t$gives to $$\psi (t)\phi (t)^{-2m-2}\geq \psi (0)\phi (0)^{-2m-2}\equiv M. \label{e4.8}$$ Combining \eqref{e4.4} with \eqref{e4.8} and integrating the resultant differential inequality, we have $$\phi (t)^{-2m-1}\leq \phi (0)^{-2m-1}-(2m+1)(m+1)Mt \label{e4.9}$$ 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. \begin{thebibliography}{99} \bibitem{b1} J. 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