\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 224, pp. 1--12.\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-2013/224\hfil Global existence and blow-up of solutions] {Global existence and blow-up of solutions for parabolic systems with nonlinear nonlocal boundary conditions} \author[Z. Sen, Z. Yang \hfil EJDE-2013/224\hfilneg] {Zhou Sen, Zuodong Yang} % in alphabetical order \address{Zhou Sen \newline Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China. \newline College of Sciences, China Pharmaceutical University, Jiangsu Nanjing 211198, China} \address{Zuodong Yang \newline Institute of Mathematics, School of Mathematical Sciences, Nanjing Normal University, Jiangsu Nanjing 210023, China} \email{zdyang\_jin@263.net} \thanks{Submitted December 12, 2012. Published October 11, 2013.} \subjclass[2000]{35K20, 35B44} \keywords{Parabolic system; nonlinear nonlocal boundary condition; \hfill\break\indent blow-up; global solution} \begin{abstract} In this article we study a nonlinear parabolic system with nonlinear nonlocal boundary conditions. We prove the uniqueness of the solutions and establish the conditions for global solutions and non-global solutions. It is interesting to observe that the weight function for the nonlocal Dirichlet boundary conditions plays a crucial role in determining whether the solutions are global or blow up in finite time. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article, we consider the parabolic system with nonlinear nonlocal boundary conditions $$\label{e1.1} \begin{gathered} u_t=\Delta u+v^p,\quad x\in\Omega,\; t>0,\\ v_t=\Delta v+u^q,\quad x\in\Omega,\; t>0,\\ u(x,t)=\int_\Omega f(x,y)u^r(y,t)\,dy,\quad x\in\partial\Omega,\; t>0,\\ v(x,t)=\int_\Omega g(x,y)v^r(y,t)\,dy,\quad x\in\partial\Omega,\; t>0, \\ u(x,0)=u_0(x), v(x,0)=v_0(x),\quad x\in\Omega, \end{gathered}$$ where $\Omega$ is a bounded domain in $R^N(N\geq1)$ with smooth boundary $\partial\Omega$, and $p, q, r>0$. The functions $f$ and $g$ are nonnegative, continuous, defined for $x\in\partial\Omega$, $y\in\Omega$ and $t\geq0$. The initial data $u_0(x)$ and $v_0(x)$ are nonnegative continuous functions satisfying the boundary conditions at $t=0$. Over the previous twenty years, many physical phenomena were formulated into nonlocal mathematical models (see \cite{AD,BB,BT,Da2,FG}). There has been a considerable amount of literature dealing with the properties of solutions to local semilinear parabolic equation or systems of heat equations with homogeneous Diriclet boundary conditions or with nonlinear boundary conditions (see \cite{CY,DLX1,DLX2,DK,LL,LX,S1,S2,P1,P2,P3,W,ZK} and references therein). However, there are some important phenomena formulated as parabolic equations which are coupled with nonlocal boundary conditions in mathematical modeling such as thermoelasticity theory (see \cite{CL,Da1,Y}). The problem of nonlocal boundary value for linear parabolic equations is of the type $$\label{e1.2} \begin{gathered} u_t-Au=c(x)u,\quad x\in\Omega,\; t>0,\\ u(x,t)=\int_\Omega K(x,y)u(y,t)\,dy,\quad x\in\partial\Omega,\; t>0, \\ u(x,0)=u_0(x), \quad x\in\Omega \end{gathered}$$ with uniformly elliptic operator $A=\sum^n_{i,j=1} a_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j} -\sum^n_{i=1} b_i(x)\frac{\partial}{\partial x_i}$ and $c(x)\leq0$ was studied by Friedman \cite{A}. The global existence and monotonic decay of the solution of problem \eqref{e1.2} was obtained under the condition $\int_\Omega |K(x,y)|\,dy<1$ for all $x\in\partial\Omega$. And later the problem \eqref{e1.2} with $Au$ replaced by $\Delta u$ and the linear term $c(x)u$ replaced by the nonlinear term $g(x,u)$ was discussed by Deng \cite{D}. The comparison principle and the local existence were established. On the basis of Deng's work, Seo \cite{S} investigated the above problem with $g(x,u)=g(u)$, by using the upper and lower solution's technique, he gained the blow-up condition of positive solution, and in the special case $g(u)=u^p$ or $g(u)=e^u$ he also derived the blow-up rate estimates. For more general discussions on the \,dynamics of parabolic problems with nonlocal boundary conditions, Pao \cite{P1} consider the problem $$\label{e1.3} \begin{gathered} u_t-Lu=f(x,u),\quad x\in\Omega,\; t>0,\\ Bu=\int_\Omega K(x,y)u(y,t)\,dy,\quad x\in\partial\Omega,\; t>0, \\ u(x,0)=u_0(x), \quad x\in\Omega \end{gathered}$$ where $Lu=\sum^n_{i,j=1}a_{ij}(x)\frac{\partial^2u}{\partial x_i\partial x_j} -\sum^n_{i=1} b_i(x)\frac{\partial u}{x_i},\quad Bu=\alpha_0\frac{\partial u}{\partial\nu}+u.$ The scalar problems with both nonlocal sources and nonlocal boundary conditions have been studied as well. For example, the problem $$\label{e1.4} \begin{gathered} u_t-\Delta u=\int_\Omega g(u)\,dy,\quad x\in\Omega,\; t>0,\\ Bu=\int_\Omega K(x,y)u(y,t)\,dy,\quad x\in\partial\Omega,\; t>0, \\ u(x,0)=u_0(x), \quad x\in\Omega \end{gathered}$$ was studied by Lin and Liu \cite{LL}, where $\int_\Omega g(u)\,dy\equiv\int_\Omega g(u(y,t))\,dy$, and Zheng and Kong \cite{ZK} established global existence condition for solution to a nonlocal parabolic system subject to nonlocal Dirichlet boundary conditions $$\label{e1.5} \begin{gathered} u_t=\Delta u-u^m(x,t)\int_\Omega v^n(y,t)\,dy,\quad x\in\Omega,\; t>0,\\ v_t=\Delta v-v^q(x,t)\int_\Omega u^p(y,t)\,dy,\quad x\in\Omega,\; t>0,\\ u=\int_\Omega \varphi(x,y)u(y,t)\,dy,\quad v=\int_\Omega \psi(x,y)v(y,t)\,dy,\quad x\in\partial\Omega,\; t>0,\\ u(x,0)=u_0(x), \quad v(x,0)=v_0(x),\quad x\in\Omega. \end{gathered}$$ Recently, Gladkov and Kim \cite{GK1} studied the heat equation with nonlinear nonlocal boundary condition, $$\label{e1.6} \begin{gathered} u_t=\Delta u+c(x,t)u^p,\quad x\in\Omega,\; t>0,\\ u(x,t)=\int_\Omega k(x,y,t)u^l(y,t)\,dy,\quad x\in\partial\Omega,\; t>0, \\ u(x,0)=u_0(x), \quad x\in\Omega\,. \end{gathered}$$ The comparison principle, the uniqueness of solution with any initial data for $\min(p,l)\geq1$ and with nontrivial initial data otherwise, non-uniqueness of solution with trivial initial data for $p<1$ or $l<1$, and local existence theorem had been proved. And in \cite{GK2} they presented some criteria for the existence of global behavior of the coefficients $c(x,t)$ and $k(x,y,t)$ as $t$ tends to infinity. Motivated by the above works, we are interested in the blow-up properties of problem \eqref{e1.1}. The aim of this paper is to establish the global existence and finite time blow-up conditions for the solution of problem \eqref{e1.1}. Before stating our main results, we state the following assumptions on the kernels $f(x,y)$, $g(x,y)$ and the initial data $u_0(x)$, $v_0(x)$: \begin{itemize} \item[(H1)] $f(x,y)$ and $g(x,y)$ are continuous and nonnegative functions on $\partial\Omega\times\bar\Omega$ and satisfy $$\int_\Omega f(x,y)\,dy>0,\quad \int_\Omega g(x,y)\,dy>0\quad \text{for all } x\in\partial\Omega.$$ \item[(H2)] $u_0(x), v_0(x)\in C^{2+\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$, $u_0(x)>0$ and $v_0(x)>0$ in $\Omega$ satisfy $$u_0(x)=\int_\Omega f(x,y)u_0^r(y)\,dy,\quad v_0(x)=\int_\Omega g(x,y)v_0^r(y)\,dy \quad \text{on } \partial\Omega.$$ \end{itemize} This article is organized as follows. Section 2 is devoted to dealing with the comparison principle and the local existence in time for problem \eqref{e1.1}. In Section 3 we give the global existence for $p,q\leq 1$. The blow-up conditions for $p,q>1$ with large initial data will be established in Section 4. In Section 5, we discuss the blow-up solutions and the existence of global solutions for $p>1>q$ or $q>1> p$. \section{The comparison principle and existence of local solutions} In this section we start with the definition of supersolution and subsolution of problem \eqref{e1.1}. Then we present some material needed in the proof of our main results. For convenience, We set $Q_T=\Omega\times(0,T]$, $S_T=\partial\Omega\times(0,T]$, $Q_t=\Omega\times(0,t]$, $00$, $\hat{v}>0$ in $\overline{Q}_T$ for all $t>0$. \end{lemma} \begin{proof} Since $u_0$ is a nontrivial nonnegative function and $\hat{u}_t-\Delta\hat{u}\geq \hat{v}^p\geq0$, a minimum of $u$ over $\bar{Q}_T$ should be attained only at a parabolic boundary point by the strong maximum principle. Thus, $\hat{u}(x,t)>0$ in $\Omega\times(0,T]$. By \eqref{e1.1} and (H1), we have $\hat{u}(x,t)>0$ for $x\in\partial\Omega$, $00$ in $\overline{Q}_T$ for all $t>0$ can be proved. \end{proof} \begin{lemma} \label{lem2.4} Let $(\hat{u},\hat{v})$ and $(\tilde{u},\tilde{v})$ be a nonnegative supersolution and a nonnegative subsolution of \eqref{e1.1} in $Q_T$, respectively, and $\hat{u}(x,0)>\tilde{u}(x,0), \hat{v}(x,0)>\tilde{v}(x,0)$ in $\overline{\Omega}$. Suppose {\rm (H1)} holds or $(\hat{u},\hat{v})$ is a strict supersolution. Then $\hat{u}>\tilde{u}$ and $\hat{v}>\tilde{v}$ hold in $\overline{Q}_T$. \end{lemma} \begin{proof} Set $\varphi(x,t)=\hat{u}-\tilde{u}$, $\psi=\hat{v}-\tilde{v}$, then $\varphi, \psi$ satisfy \label{e2.2} \begin{aligned} \varphi_t-\Delta\varphi\geq{\hat{v}}^p-{\tilde{v}}^q\geq\rho_1(\theta_v)\psi, \quad (x,t)\in Q_T,\\ \psi_t-\Delta\psi\geq{\hat{u}}^p-{\tilde{u}}^q\geq\rho_2(\theta_u)\varphi, \quad (x,t)\in Q_T, \end{aligned} with the following boundary and initial conditions \begin{gather} \label{e2.3} \begin{gathered} \varphi(x,t)\geq\int_\Omega f(x,y)h_1(\theta_u)\varphi(y,t)\,dy,\quad (x,t)\in S_T,\\ \psi(x,t)\geq\int_\Omega g(x,y)h_2(\theta_v)\psi(y,t)\,dy,\quad (x,t)\in S_T, \end{gathered} \\ \label{e2.4} \varphi(x,0)>0,\quad \psi(x,0)>0,\quad x\in\bar\Omega, \end{gather} where $\theta_u$ is between $\hat{u}$ and $\tilde{u}$, and $\theta_v$ is between $\hat{v}$ and $\tilde{v}$. Since $\varphi(x,0), \psi(x,0)>0$, by continuity, there exists $\delta>0$ such that $\varphi, \psi>0$ for $(x,t)\in\bar{\Omega}\times(0,\delta)$. Suppose for a contradiction that t_0=\sup\{t\in(0,T), \varphi, \psi>0 \text{on}~\bar{\Omega}\times(0,t)\}0 $$with boundary conditions$$ u=0,\quad x\in\partial\Omega,\; t>0. Then for y\in\partial\Omega, G(x,y;t)=0 and \frac{\partial G(x,y;t)}{\partial n}<0. Applying G(x,y;t) to \eqref{e2.2}, we have \begin{align*} \varphi(x,t) &\geq\int_\Omega G(x,y;t)\varphi(y,0)\,dy\\ &\quad +\int_0^t\int_\Omega G(x,y;t-\eta)\rho_1(\theta_v)\psi(y,\eta)\,dy\,d\eta\\ &\quad -\int_0^t\int_{\partial\Omega} \frac{\partial G(x,\xi;t-\eta)}{\partial n} \int_\Omega f(\xi,y)h_1(\theta_u)\varphi(y,\eta)\,dy\,d\xi \,d\eta. \end{align*} Since \varphi, \psi>0 for all x\in\bar{\Omega}, 00, we find that \varphi(x,t_0)\geq\int_\Omega G(x,y;t)\varphi(y,0)\,dy>0. $$In particular, \varphi(x_0,t_0)>0, which contradicts our previous assumption. \end{proof} \begin{remark} \label{rmk2.5} \rm If \int_\Omega f(x,y)\,dy\leq1, \int_\Omega g(x,y)\,dy\leq1, we need only \hat{u}(x,0)\geq\tilde{u}(x,0), \hat{v}(x,0)\geq\tilde{v}(x,0) in \bar{\Omega}, since for any \varepsilon>0, \varphi(x,t)=\hat{u}+\varepsilon-\tilde{u}, \psi=\hat{v}+\varepsilon-\tilde{v} satisfy all inequalities in \eqref{e2.2}-\eqref{e2.4}, then we have \hat{u}+\varepsilon>\tilde{u}, \hat{v}+\varepsilon>\tilde{v}, and it follows that \hat{u}\geq\tilde{u}, \hat{v}\geq\tilde{v}. \end{remark} Let {\varepsilon_m} be decreasing to 0 such that 0<\varepsilon_m<1. For \varepsilon=\varepsilon_m, let u_{0\varepsilon},v_{0\varepsilon} be the functions with the following properties: u_{0\varepsilon}, v_{0\varepsilon}\in C(\bar\Omega), u_{0\varepsilon}>\varepsilon, v_{0\varepsilon}>\varepsilon, u_{0\varepsilon_i}>u_{0\varepsilon_j}, v_{0\varepsilon_i}>v_{0\varepsilon_j} for \varepsilon_i>\varepsilon_j, u_{0\varepsilon}\to u_0(x), v_{0\varepsilon}\to v_0(x) as \varepsilon\to 0, and u_{0\varepsilon}(x)=\int_{\Omega}f(x,y)u^r_{0\varepsilon}(y)\,dy, v_{0\varepsilon}(x)=\int_{\Omega}g(x,y)v^r_{0\varepsilon}(y)\,dy. Since the nonlinearities in \eqref{e1.1} do not satisfy the Lipschitz condition, we need to consider the following auxiliary problem: $$\label{e2.5} \begin{gathered} u_t=\Delta u+v^p,\quad x\in\Omega,\; t>0,\\ v_t=\Delta v+u^q,\quad x\in\Omega,\; t>0,\\ u(x,t)=\int_\Omega f(x,y)u^r(y,t)\,dy+\varepsilon,\quad x\in\partial\Omega,\; t>0,\\ v(x,t)=\int_\Omega g(x,y)v^r(y,t)\,dy+\varepsilon,\quad x\in\partial\Omega,\; t>0, \\ u(x,0)=u_0(x), \quad v(x,0)=v_0(x),\quad x\in\Omega, \end{gathered}$$ where \varepsilon=\varepsilon_m. The notion of a solution u_{\varepsilon} for problem \eqref{e2.5} can be defined in a similar way as in Definition \ref{def2.1}. \begin{theorem} \label{thm2.6} There exists T^* (0\varepsilon_1. Obviously, u_{\varepsilon_2(x,t)},v_{\varepsilon_2}(x,t) is a pair of strict supersolution of \eqref{e2.5} with \varepsilon=\varepsilon_1. By the comparison principle for problem \eqref{e2.5} we have that u_{\varepsilon_1}0$$ with a boundary conditionu=0$for$x\in\partial\Omega$,$t>0. Then from \eqref{e1.1} we obtain \begin{align*} u_m(x,t)-u(x,t) &=\int^t_0\int_\Omega G(x,y;t-\eta)\rho_1(\theta_v)(v_m-v)\,dy\,d\eta\\ &\quad -\int_0^t\int_{\partial\Omega} \frac{\partial G(x,\xi;t-\eta)}{\partial n} \int_\Omega f(\xi,y)h_1(\theta_u)(u_m-u)\,dy\,d\xi\, \,d\eta, \end{align*} where\rho_1(\theta_v)$and$h_1(\theta_u)$are continuous functions in$\bar{Q}_T. Due to the assumptions of in this theorem and Lemma \ref{lem2.3}, we have $$u_m(x,t)-u(x,t)\leq\sigma_1(T)\{\sup_{Q_T}(v_m(x,t)-v(x,t)) +\sup_{Q_T}(u_m(x,t)-u(x,t))\},$$ where \begin{align*} \sigma_1(T) &=\sup_{Q_T}\int^t_0\int_\Omega G(x,y;t-\eta)\rho_1(\theta_v)\,dy\,d\eta\\ &\quad +\sup_{Q_T}\int_0^t\int_{\partial\Omega} -\frac{\partial G(x,\xi;t-\eta)}{\partial n} \int_\Omega f(\xi,y)h_1(\theta_u)\,dy\,d\xi\,d\eta. \end{align*} Similarly we can prove that $$v_m(x,t)-v(x,t)\leq\sigma_2(T)\{\sup_{Q_T}(u_m(x,t)-u(x,t)) +\sup_{Q_T}(v_m(x,t)-v(x,t))\}.$$ ChoosingT$so small that$\sigma_1(T)+\sigma_2(T)<1$, we prove the uniqueness of solution for \eqref{e1.1} in$Q_T$. Case 2:$p,q<1$or$r<1$. We distinguish three cases:$r<1$and$p,q\leq1$;$r<1$and$p,q>1$; or$r\geq1$and$p,q<1$. To prove the uniqueness it suffices to show that if$(u,v)$is any solution of \eqref{e1.1}, then $$\label{e2.6} u(x,t)\geq u_M(x,t),\quad v(x,t)\geq v_M(x,t).$$ Let$r<1$and$p,q\leq1$. Set$w=u_M-u, z=v_M-v$. Since$r<1$and$p,q\leq1$, it is easy to verify that \begin{gather*} w_t\leq\Delta w+z^p,\quad (x,t)\in Q_{T_1},\\ z_t\leq\Delta z+w^q,\quad (x,t)\in Q_{T_1},\\ w(x,t)\leq\int_\Omega f(x,y)w^r(y,t)\,dy,\quad (x,t)\in S_{T_1},\\ z(x,t)\leq\int_\Omega g(x,y)z^r(y,t)\,dy,\quad (x,t)\in S_{T_1},\\ w(x,0)\equiv0,\quad z(x,0)\equiv0,\quad x\in\Omega. \end{gather*} By Lemma \ref{lem2.9} there exists a unique solution$w^0(x,t)>0,z^0(x,t)$for$x\in\bar\Omega$,$00, b(x,t)>0$in$Q_T$. Thus by the comparison principle we conclude that$a(x,t)\geq w^0(x,t)$,$b(x,t)\geq z^0(x,t)$. This implies \eqref{e2.6} and the theorem holds under the assumptions$r<1$and$p,q\leq1$. For the other cases we can discuss in a similar way. \end{proof} \section{Existence of global solutions for$p,q\leq1$} \begin{theorem} \label{thm3.1} Let$p,q\leq1$. Then for all$r\leq1$, the solution$(u, v)$of \eqref{e1.1} exists globally for any nonnegative initial data. \end{theorem} \begin{proof} We first suppose$01$such that$f(x,y)\leq M$and$g(x,y)\leq M$for all$(x,y)\in\partial\Omega\times\bar\Omega$. It is easy to see that the pair of functions$(\hat{u}, \hat{v})=(Ce^{\beta t}, Ce^{\beta t})$is a strict supersolution of \eqref{e1.1} if$\beta\geq M$and$C\geq\max\{\sup_{\overline{\Omega}}u_0(x), \sup_{\overline{\Omega}}v_0(x),(M|\Omega|)^{\frac{1}{1-r}}\}$. In the case$r=1$, the pair of function$(\hat{u}, \hat{v})=(Ce^{\beta t}, Ce^{\beta t})$is a strict supersolution of \eqref{e1.1} if and only if $$\int_\Omega f(x,y)\,dy<1, \int_\Omega g(x,y)\,dy<1\quad \text{for all } x\in\partial\Omega.\label{e3.1}$$ Therefore, when \eqref{e3.1} is not valid, we need to construct another supersolution. Denote by$\varphi(x)$the eigenfunction corresponding to the first eigenvalue$\lambda_1^\Omega$of the elliptic problem $$-\Delta\varphi=\lambda\varphi,\quad x\in\Omega;\quad \varphi=0,\quad x\in\partial\Omega.\label{e3.2}$$ and that for$0<\varepsilon<1$satisfies $$M\int_\Omega\frac{1}{\varphi(y)+\varepsilon}\,dy\leq1.$$ Now we set$\hat{u}=\frac{Ce^{\gamma t}}{\varphi(x)+\varepsilon}, \hat{v}=\frac{Ce^{\gamma t}}{\varphi(x)+\varepsilon}$. A simple computation shows $$\label{e3.3} \begin{gathered} \hat{u}_t-\Delta\hat{u}-\hat{v}^p \geq\gamma\hat{u}-\hat{u} \Big(\frac{\lambda_1^\Omega\varphi}{\varphi(x)+\varepsilon} +\frac{2|\nabla\varphi|^2}{(\varphi(x)+\varepsilon)^2}\Big)-\hat{v}\geq0,\\ \hat{v}_t-\Delta\hat{v}-\hat{u}^q \geq\gamma\hat{v}-\hat{v} \Big(\frac{\lambda_1^\Omega\varphi}{\varphi(x)+\varepsilon} +\frac{2|\nabla\varphi|^2}{(\varphi(x)+\varepsilon)^2}\Big)-\hat{u}\geq0, \end{gathered}$$ if we choose $$C\geq\max\{\sup_{\overline{\Omega}}(\varphi(x)+\varepsilon), \sup_{\overline{\Omega}}u_0(x)\sup_{\overline{\Omega}}(\varphi(x)+\varepsilon), \sup_{\overline{\Omega}}v_0(x)\sup_{\overline{\Omega}}(\varphi(x)+\varepsilon)\},$$ and $$\gamma\geq\lambda_1^\Omega+ \sup_{\overline{\Omega}}\frac{2|\nabla\varphi|^2}{(\varphi+\varepsilon)^2}+1.$$ It is clear from \eqref{e3.3} and the choice of$C, \gamma$that$(\hat{u}, \hat{v})$is a strict supersolution of \eqref{e1.1}. Thus the solution of problem \eqref{e1.1} exists globally. \end{proof} \section{Blow-up in finite time for$p,q>1$} In this section we will get some blow-up results for \eqref{e1.1}. First, we give the following lemma which is crucial in proving the blow-up results. \begin{lemma} \label{lem4.1} There exists a positive solution$\phi$for the elliptic eigenvalue problem $$-\Delta\phi=\lambda\phi,\quad x\in\Omega,\quad \min_{x\in\overline{\Omega}}\phi=1. \label{e4.1}$$ \end{lemma} \begin{proof} Since$\Omega$is a bounded domain in$R^N(N\geq1)$with smooth boundary$\partial\Omega$, we can choose another bounded domain$\Omega_1$such that$\Omega\subset\subset\Omega_1$. Let$\lambda_1^{\Omega_1}$be the first eigenvalue of following elliptic eigenvalue problem $$-\Delta\phi=\lambda\phi,\quad x\in\Omega_1,\quad \phi=0, \quad x\in\partial\Omega_1$$ and$\phi_1$is the corresponding eigenfunction with$\phi_1>0$. Then$\phi_1\geq\delta>0$in$\overline{\Omega}$. Set$\phi=\frac{1}{\delta}\phi_1$, then$\phi$is the function satisfying \eqref{e4.1}. \end{proof} \begin{remark} \label{rmk4.2} \rm From the continuity of eigenvalue to the domain$\Omega$, we can choose$\Omega_1$such that$\lambda_1^\Omega>\lambda_1^{\Omega_1}>\lambda_1^\Omega-\varepsilon$for some constant$\varepsilon$, sufficiently small. \end{remark} Now we turn to the blow-up conclusions. We denote $$K=\max_{x\in\overline{\Omega}}\phi(x),\quad h_0=\min\{\min_{x\in\partial\Omega}\int_\Omega f(x,y)\,dy, \min_{x\in\partial\Omega}\int_\Omega g(x,y)\,dy\}.$$ From Lemma \ref{lem4.1} and the assumption (H1) we can see$K>1$and$h_0>0$. \begin{theorem} \label{thm4.3} Let$p,q>1$. Then if$r>1$and$h_0|\Omega|>K^{\min\{p,q\}}$, the solution$(u, v)$of \eqref{e1.1} blows up in finite time for the large initial data. \end{theorem} \begin{proof} Without lose of generality, we can assume that$p\geq q$. Set$\tilde{u}=s^l(t)\phi^q(x), \tilde{v}=s^l(t)\phi^q(x)$, where$l$is a constant and satisfies$p\geq q>l$,$r>l$. The function$\phi$is defined as in Lemma \ref{lem4.1}. Let$s(t)$be the solution to the problem $$s'(t)=-\lambda_1^{\Omega_1}qs(t)+s^l(t),\quad t>0,\quad s(0)=s_0. \label{e4.2}$$ with initial data$s_0\geq\max\{(\lambda_1^{\Omega_1}q)^{\frac{1}{l-1}}, 1\}$. It is easy to see that$s(t)\geq1$and blows up in finite time$T_{s_0}$. A direct computation yields $$\label{e4.3} \begin{split} \tilde{u}_t-\Delta\tilde{u}-\tilde{v}^p &=ls^{l-1}(t)s'(t)\phi^q-s^l(t)[-\lambda_1q\phi^q+q(q-1)\phi^{q-2}|\nabla\phi|^2] -s^{lp}(t)\phi^{pq}\\ &\leq ls^{l-1}s'(t)\phi^q+s^l(t)\lambda_1q\phi^q-s^l(t)\phi^q\\ &=ls^{l-1}(t)(s'(t)+\lambda_1qs(t)-s^l(t))\phi^q=0,\\ \end{split}$$ $$\label{e4.4} \begin{split} \tilde{v}_t-\Delta\tilde{v}-\tilde{u}^q &=ls^{\alpha-1}s'(t)\phi^q-s^l(t)[-\lambda_1q\phi^q+q(q-1)\phi^{q-2}|\nabla\phi|^2] -s^{lq}(t)\phi^{q^2}\\ &\leq ls^{l-1}s'(t)\phi^q+s^l(t)\lambda_1q\phi^q-s^l\phi^q\\ &=ls^{l-1}(t)(s'(t)+\lambda_1qs(t)-s^l(t))\phi^q=0, \end{split}$$ in$\Omega\times(0,T_{s_0})$, and for$x\in\partial\Omega\times(0,T_{s_0})$. So we have $$\label{e4.5} \begin{split} \tilde{u}(x,t)&\leq s^l(t)K^q\leq h_0s^l(t)|\Omega|\\ &\leq\int_\Omega f(x,y)s^{lr}(t)\phi^{qr}(y)\,dy\\ &=\int_\Omega f(x,y)\tilde{u}^r(y,t)\,dy,\\ \tilde{v}(x,t)&\leq s^l(t)K^q\leq h_0s^l(t)|\Omega|\\ &\leq\int_\Omega g(x,y)s^{lr}(t)\phi^{qr}(y)\,dy\\ &=\int_\Omega g(x,y)\tilde{v}^r(y,t)\,dy. \end{split}$$ From \eqref{e4.3}-\eqref{e4.5} we can see that$(\tilde{u},\tilde{v})$is a subsolution provided the initial data so large that$s^l(0)\phi^q(x)\leq u_0(x), s^l(0)\phi^q(x)\leq v_0(x)$for$x\in\Omega$. Thus by Lemma \ref{lem2.4}, the solution$(u,v)$of problem \eqref{e1.1} blows up because$(\tilde{u},\tilde{v})$blows up in finite time. \end{proof} \section{Blow-up and global solution for$p>1>q$or$q>1>p$} \begin{theorem} \label{thm5.1} Let$p>1>q$or$q>1>p$. If$pq<1$,$r\leq1$, and $$\int_\Omega f(x,y)\,dy\leq1, \quad \int_\Omega g(x,y)\,dy\leq 1$$ for all$x\in\partial\Omega$, then the solution$(u, v)$of \eqref{e1.1} exists globally for sufficiently small initial data. \end{theorem} \begin{proof} We assume$p>1>q$. Since$pq<1$, there exists a constant$l$such that$00,\quad s(0)=s_0 $$with initial data s_0>1. It is easy to see that s(t)>1 and exists globally. Set \hat{u}=s^{p+1}(t) and \hat{v}=s^{q+1}(t). Then we have $$\label{e5.1} \begin{split} \hat{u}_t-\Delta\hat{u}-\hat{v}^p &=(p+1)s^{p}(t)s'(t)-s^{(q+1)p}(t)\\ &=(p+1)s^{p}(t)s^l(t)-s^{(q+1)p}(t)\\ &>s^{p+l}(t)-s^{(q+1)p}(t)\geq0, \end{split}$$ $$\label{e5.2} \begin{split} \hat{v}_t-\Delta\hat{v}-\hat{u}^q &=(q+1)s^{q}(t)s'(t)-s^{(p+1)q}(t)\\ &=(q+1)s^{q}(t)s^l(t)-s^{(p+1)q}(t)\\ &>s^{q+l}(t)-s^{(p+1)q}(t)\geq0, \end{split}$$ in \Omega\times(0,\infty), and for x\in\partial\Omega\times(0,\infty). So we have $$\label{e5.3} \begin{gathered} \hat{u}=s^{p+1}(t)\geq\int_\Omega f(x,y)s^{(p+1)r}(t)\,dy =\int_\Omega f(x,y)\hat{u}^r\,dy,\\ \hat{v}=s^{q+1}(t)\geq\int_\Omega g(x,y)s^{(q+1)r}(t)\,dy =\int_\Omega g(x,y)\hat{v}^r\,dy. \end{gathered}$$ From \eqref{e5.1}--\eqref{e5.3} we see that (\hat{u},\hat{v}) is a supersolution provided that s_0^{p+1}\geq u_0(x), s_0^{q+1}\geq v_0(x) for x\in\Omega. The case q>1>p can be treated by exchanging the roles of u and v in the above case. \end{proof} \begin{theorem} \label{thm5.2} Let p>1>q or q>1>p. If pq>1, r\geq1, and$$ \int_\Omega f(x,y)\,dy\geq1, \int_\Omega g(x,y)\,dy\geq1 $$for all x\in\partial\Omega, then the solution (u, v) of \eqref{e1.1} blows up in finite time for sufficiently large initial data. \end{theorem} \begin{proof} We assume p>1>q. Since pq>1, there exists a constant l such that 10,\quad s(0)=s_0$$ with the initial data $s_0>1$. It is easy to see that $s(t)>1$ and blows up in finite time. Set $\tilde{u}=s^{p+1}(t),\quad \tilde{v}=s^{q+1}(t)$, then we have \label{e5.4} \begin{split} \tilde{u}_t-\Delta\tilde{u}-\tilde{v}^p &=(p+1)s^{p}(t)s'(t)-s^{(q+1)p}(t)\\ &=(p+1)s^{p}(t)s^l(t)-s^{(q+1)p}(t)\\ &1>p$can be treated by exchanging the roles of$u$and$v\$ in the above case. The proof is complete. \end{proof} \subsection*{Acknowledgments} This work is Supported by the National Natural Science Foundation of China (grant 11171092), the Natural Science Foundation of Educational Department of Jiangsu Province (grant 08KJB110005), and partially supported by the Project of Graduate Education Innovation of Jiangsu Province (grant CXLX12\_0389). \begin{thebibliography}{00} \bibitem{AD} J. R. Anderson, K. Deng; \emph{Global existence for degenerate parabolic equations with a nonlocal forcing}, Math. Methods Appl. Sci. \textbf{20} (1997), 1069-1087. \bibitem{BB} J. W. Bebernes, A. Bressan; \emph{Thermal behaviour for a confined reactive gas}, J. 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