\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
\begin{equation} \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}
\end{equation}
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
\begin{equation}\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}
\end{equation}
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
\begin{equation}\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}
\end{equation}
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
\begin{equation} \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}
\end{equation}
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
\begin{equation}\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}
\end{equation}

Recently, Gladkov and Kim \cite{GK1} studied the heat equation with
 nonlinear nonlocal boundary condition,
\begin{equation}\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}
\end{equation}
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
$Q_T=\Omega\times(0,T]$, $S_T=\partial\Omega\times(0,T]$,
$Q_t=\Omega\times(0,t]$, $0<t\leq T<\infty$,
where $\overline{Q}_T$ and $\overline{Q}_t$ are their respective closures.

\begin{definition} \label{def2.1} \rm
A pair of nonnegative functions
$\tilde{u},\tilde{v}\in C^{2,1}(Q_T)\cap C(\overline{Q}_T)$
is called a subsolution of \eqref{e1.1} if
\begin{equation} \label{e2.1}
\begin{gathered}
\tilde{u}_t\leq\Delta \tilde{u}+\tilde{v}^p,\quad (x,t)\in Q_T,\\
\tilde{v}_t\leq\Delta \tilde{v}+\tilde{u}^q,\quad (x,t)\in Q_T,\\
\tilde{u}(x,t)\leq\int_\Omega f(x,y)\tilde{u}^r(y,t)\,dy,\quad (x,t)\in S_T,\\
\tilde{v}(x,t)\leq\int_\Omega g(x,y)\tilde{v}^r(y,t)\,dy,\quad (x,t)\in S_T,\\
\tilde{u}(x,0)\leq u_0(x), \tilde{v}\leq v_0(x),\quad x\in\Omega.
\end{gathered}
\end{equation}
and a pair of functions $\hat{u},\hat{v}\in C^{2,1}(Q_T)\cap C(\bar{Q}_T)$
is a supersolution of problem \eqref{e1.1} if $\hat{u},\hat{v}\geq0$ and
it satisfies inequalities in \eqref{e2.1} in the reverse order. Furthermore, we
say that $u$ and $v$ are solutions of problem \eqref{e1.1} in $\bar{Q}_T$ if they
are both subsolutions and supersolutions of \eqref{e1.1} in $\bar{Q}_T$.
\end{definition}

\begin{definition} \label{def2.2} \rm
We say a pair of nonnegative functions $\tilde{u},\tilde{v}$
is a strict subsolution of  \eqref{e1.1} in $Q_T$ if it is a
subsolution and the equalities in boundary conditions in \eqref{e2.1}
is strict. Similarly, a strict supersolution is defined by the
opposite inequalities.
\end{definition}

The following lemma and comparison principle play a crucial role
 in our discussions.

\begin{lemma}\label{lem2.3}
Let $u_0, v_0$ be nontrivial functions in $\overline{\Omega}$ and
assume that the assumptions {\rm (H1)--(H2)} hold. Suppose that
$(\hat{u},\hat{v})$ is a supersolution of \eqref{e1.1} in $Q_T$, then
$\hat{u}>0$, $\hat{v}>0$ 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 $x\in\partial\Omega$, $0<t\leq T$. Similarly,
$\hat{v}(x,t)>0$ 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
\begin{equation} \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}
\end{equation}
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
$t_0=\sup\{t\in(0,T), \varphi, \psi>0 \text{on}~\bar{\Omega}\times(0,t)\}<T$.
Then $\varphi, \psi\geq0$ on $\bar{Q}_{\bar{t}_0}$, and at least one of
$\varphi, \psi$ vanishes at $(x_0,t_0)$ for some $x_0\in\bar{\Omega}$.
Without loss of generality, we suppose $\varphi(x_0,t_0)=0$. Let $G(x,y;t)$
denote the Green's function for
$$
Lu=u_t-\Delta u,\quad x\in\Omega,\ t>0
$$
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 $x\in\bar{\Omega}$, $0<t<t_0$, and
$f(x,y)>0$, 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:
\begin{equation} \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}
\end{equation}
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 $T^*$ $(0<T^*\leq\infty)$ such that  \eqref{e2.5} has a unique
solution $(u(x,t), v(x,t))\in C(\bar{\Omega}\times[0,T^*))\cap C^{2,1}
(\Omega\times(0,T^*))$.
\end{theorem}

Let $\varepsilon_2>\varepsilon_1$. 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
$u_{\varepsilon_1}<u_{\varepsilon_2}$, $v_{\varepsilon_1}<v_{\varepsilon_2}$.
Taking $\varepsilon\to 0$, we get
\[
u_M(x,t)=\lim_{\varepsilon\to 0}u_{\varepsilon}(x,t),\quad
v_M(x,t)=\lim_{\varepsilon\to 0}v_{\varepsilon}(x,t)
\]
with $u_M(x,t)\geq0, v_M(x,t)\geq0$. It is easy to check that
$(u_M(x,t), v_M(x,t))$ are solutions of \eqref{e1.1}.
 Let $\bar{u},\bar{v}$
be any solution of \eqref{e1.1}, then by comparison principle
$u_{\varepsilon}\geq\bar{u}$, $v_{\varepsilon}\geq\bar{v}$. Taking
$\varepsilon\to 0$, we conclude that $u_M\geq\bar{u}$, $v_M\geq\bar{v}$.
So we have the existence of a local solution.

\begin{theorem} \label{thm2.7}
There exists $T$ $(0<T\leq\infty)$ such that  \eqref{e1.1} has a
maximal solution $(u(x,t), v(x,t))\in C(\bar{\Omega}\times[0,T))
\cap C^{2,1}(\Omega\times(0,T))$.
\end{theorem}

We now give the uniqueness for solutions to \eqref{e1.1}.

\begin{theorem} \label{thm2.8}
Assume that {\rm (H1), (H2)} hold. If $p,q,r\geq1$, or if $p,q<1$ or $r<1$,
then \eqref{e1.1} has a unique solution
$(u(x,t), v(x,t))\in C(\bar{\Omega}\times[0,T))\cap C^{2,1}(\Omega\times(0,T))$.
\end{theorem}

\begin{lemma} \label{lem2.9}
Assume {\rm (H1)} holds, $\min\{\max\{p,q\},r\}\leq 1$, and
$u_0(x)\equiv0$, $v_0(x)\equiv0$. Then maximal solution $u_M, v_M$ of 
\eqref{e1.1} is strictly positive for $x\in\bar{\Omega}$ and all positive
time, as long as it exists.
\end{lemma}


\begin{proof}[Proof of Theorem \ref{thm2.8}]
Case 1:  $p,q,r\geq 1$.
Assume that \eqref{e1.1} has the maximal solution $(u_M(x,t),v_M(x,t))$
and another solution $(u(x,t),v(x,t))$. Then there exists $t_0\geq0$,
such that $(u_M(x,t),v_M(x,t))\equiv(u(x,t),v(x,t))$ for $x\in\bar{\Omega}$,
$0\leq t\leq t_0$ and $(u_M(x,t),v_M(x,t))\not\equiv(u(x,t),v(x,t))$ for
$x\in\bar{\Omega}$, $t_0\leq t\leq t_0+\gamma$ with $\gamma\in(0,T-t_0)$.
We can assume that $t_0=0$. Let $G(x,y;t)$ denote the Green's function
for the heat equation
$$
u_t-\Delta u=0,\quad x\in\Omega,\; t>0
$$
with a boundary condition $u=0$ 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
$$
v_m(x,t)-v(x,t)\leq\sigma_2(T)\{\sup_{Q_T}(u_m(x,t)-u(x,t))
+\sup_{Q_T}(v_m(x,t)-v(x,t))\}.
$$
Choosing $T$ 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
\begin{equation} \label{e2.6}
u(x,t)\geq u_M(x,t),\quad v(x,t)\geq v_M(x,t).
\end{equation}
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
$x\in\bar\Omega$, $0<t<T_2$ satisfying the equations in \eqref{e1.1} and the
boundary conditions
$$
w^0(x,0)=0,\quad z^0(x,0)=0.
$$
In a similar way as that used in Lemma \ref{lem2.4}, we can prove that 
$w^0(x,t)\geq w(x,t)$,
$z^0(x,t)\geq z(x,t)$ and $u_M(x,t)\geq w^0(x,t)$, $v_M(x,t)\geq z^0(x,t)$.
We put $a(x,t)=w^0(x,t)-w(x,t)$, $b(x,t)=z^0(x,t)-z(x,t)$ and obtain
\begin{gather*}
a_t\geq\Delta a+b^p,\quad (x,t)\in Q_{T_3},\\
b_t\geq\Delta b+a^q,\quad (x,t)\in Q_{T_3},\\
a(x,t)\geq\int_\Omega f(x,y)a^r(y,t)\,dy,\quad (x,t)\in S_{T_3},\\
b(x,t)\geq\int_\Omega g(x,y)b^r(y,t)\,dy,\quad (x,t)\in S_{T_3},\\
a(x,0)\equiv0,\quad b(x,0)\equiv0,\quad x\in\Omega.
\end{gather*}
It is not difficult to prove that $a(x,t)>0, b(x,t)>0$ 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 $0<r<1$. By the conditions for
$f(x,y), g(x,y)$, there exists a constant $M>1$ 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
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
-\Delta\varphi=\lambda\varphi,\quad x\in\Omega;\quad
\varphi=0,\quad x\in\partial\Omega.\label{e3.2}
\end{equation}
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
\begin{equation} \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}
\end{equation}
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
\begin{equation}
-\Delta\phi=\lambda\phi,\quad x\in\Omega,\quad
\min_{x\in\overline{\Omega}}\phi=1. \label{e4.1}
\end{equation}
\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
\begin{equation}
s'(t)=-\lambda_1^{\Omega_1}qs(t)+s^l(t),\quad  t>0,\quad s(0)=s_0. \label{e4.2}
\end{equation}
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
\begin{equation} \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}
\end{equation}
\begin{equation} \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}
\end{equation}
in $\Omega\times(0,T_{s_0})$, and for $x\in\partial\Omega\times(0,T_{s_0})$. So
we have
\begin{equation} \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}
\end{equation}
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 $0<pq\leq l<1$. Let $s(t)$ be the unique solution to
the  problem
$$
s'(t)=s^l(t),\quad t>0,\quad  s(0)=s_0
$$
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
\begin{equation} \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}
\end{equation}
\begin{equation} \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}
\end{equation}
in $\Omega\times(0,\infty)$, and for $x\in\partial\Omega\times(0,\infty)$. So
we have
\begin{equation} \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}
\end{equation}
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 $1<l\leq pq$. Let $s(t)$ be the unique
solution to the  problem
$$
s'(t)=s^l(t),\quad t>0,\quad s(0)=s_0
$$
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
\begin{equation} \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)\\
&<s^{p+l}(t)-s^{(q+1)p}(t)\leq0,
\end{split}
\end{equation}
\begin{equation} \label{e5.5}
\begin{split}
\tilde{v}_t-\Delta\tilde{v}-\tilde{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)\leq0,
\end{split}
\end{equation}
in $\Omega\times(0,\infty)$, and for $x\in\partial\Omega\times(0,\infty)$. So
we have
\begin{equation} \label{e5.6}
\begin{gathered}
\tilde{u}=s^{p+1}(t)\leq\int_\Omega f(x,y)s^{(p+1)r}(t)\,dy
=\int_\Omega f(x,y)\tilde{u}^r\,dy,\\
\tilde{v}=s^{q+1}(t)\leq\int_\Omega g(x,y)s^{(q+1)r}(t)\,dy
=\int_\Omega g(x,y)\tilde{v}^r\,dy.
\end{gathered}
\end{equation}
From  \eqref{e5.4}--\eqref{e5.6} we can see that
$(\tilde{u},\tilde{v})$ is a supersolution provided
$s_0^{p+1}\leq u_0(x)$, $s_0^{q+1}\leq v_0(x)$
for $x\in\Omega$. Since $(\tilde{u},\tilde{v})$ blows up in finite
time. The case $q>1>p$ 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. Differential Equations. \textbf{44} (1982), 118-133.

\bibitem{BT} J. W. Bebernes, P. Talaga;
\emph{Nonlocal problems modelling shear banding},
Nonlinear Anal. \textbf{3} (1996), 79-103.

\bibitem{CL} Y. P. Chen, L. H. Liu;
\emph{Global blow-up for a localized nonlinear
parabolic equation with a nonlocal boundary condition},
J. Math. Anal. Appl. \textbf{384} (2011), 421-430.

\bibitem{CY} Z. J. Cui, Z. D. Yang;
\emph{Roles of weight functions to a nonlinear
porous medium equation with nonlocal source and nonlocal boundary
condition}, J. Math. Anal. Appl. \textbf{342} (2008), 559-570.

\bibitem{Da1} W. A. Day;
\emph{A decreasing property of solutions of parabolic equations
with applications to thermoelasticity}, Quart. Appl. Math. \textbf{40}
(1983), 468-475.

\bibitem{Da2} W. A. Day;
\emph{Extensions of property of heat equation to linear
thermoelasticity and other theories},
Quart. Appl. Math. \textbf{40} (1982), 319-330.

\bibitem{D} K. Deng;
\emph{Comparison principle for some nonlocal problems},
Quart. Appl. Math. \textbf{50} (1992), 517-522.

\bibitem{DLX1} W. Deng, Y. Li, C. Xie;
\emph{Global existence and nonexistence for a class
of degenerate parabolic systems}, Nonlinear Anal. \textbf{55}
(2003), 233-244.

\bibitem{DLX2} W. Deng, Y. Li, C. Xie;
\emph{Existence and nonexistence of global
solutions of some nonlocal degenerate parabolic equations},
Appl. Math. Lett. \textbf{16} (2003), 803-808.

\bibitem{DK} J. I. Diaz, R. Kerser;
\emph{On a nonlinear degenerate parabolic equation
in infiltration or evaporation through a porous medium},
J. Differential equations. \textbf{69} (1987), 368-403.

\bibitem{A} A. Friedman;
\emph{Monotonic decay of solutions of parabolic equations
with nonlocal boundary condition}, Quart. Appl. Math. \textbf{44}
(1986), 401-407.

\bibitem{FG} J. Furter, M. Grinfeld;
\emph{Local vs. nonlocal interactions in population \,dynamics},
J. Math. Biol. \textbf{27} (1989). 65-80.

\bibitem{GK1} A. L. Gladkov, K. I. Kim;
\emph{Uniqueness and nonuniqueness for
reaction-diffusion equation with nonlocal boundary condition},
Adv. Math. Sci. Appl. \textbf{19} (2009), 39-49.

\bibitem{GK2} A. L. Gladkov, K. I. Kim;
\emph{Blow-up of solutions for semilinear heat
equation with nonlinear nonlocal boundary condition},
J. Math. Anal. Appl. \textbf{338} (2008), 264-273.

\bibitem{LL} Z. G. Lin, Y. R. Liu;
\emph{Uniform blow-up profiles for diffusion equations
with nonlocal source and nonlocal boundary}, Acta. Math. Sci. Ser.
B \textbf{24} (2004), 443-450.

\bibitem{LX} Z. G. Lin, C. H. Xie;
\emph{The blow-up rate for a system of heat equations
with nonlinear boundary condition}, Nonlinear Anal. \textbf{34}
(1998) 767-778.

\bibitem{S1} P. Souplet;
\emph{Uniform blow-up profiles and boundary behavior for
diffusion equations with nonlocal nonlinear source},
J. Differential equations. \textbf{153} (1999), 374-406.

\bibitem{S2} P. Souplet;
\emph{Blow-up in nonlocal reaction-diffusion equations},
SIAM J. Math. Anal. \textbf{29} (1998), 1301-1334.

\bibitem{P1} C. V. Pao;
\emph{Numerical solutions of reaction-diffusion equations with
nonlocal boundary conditions}, J. Comput. Appl. Math. \textbf{136}
(2001), 227-243.

\bibitem{P2} C. V. Pao;
\emph{Asymptotic behavior of solutions of reaction-diffusion
equations with nonlocal boundary conditions}, J. Comput. Appl. Math.
\textbf{88} (1998), 767-778.

\bibitem{P3} C. V. Pao;
\emph{Dynamics of reaction-diffusion equations with nonlocal
boundary conditions}, Quart. Appl. Math. \textbf{50} (1995), 173-186.

\bibitem{S} S. Seo;
\emph{Blowup of solutions to heat equations with nonlocal
boundary conditions}, Kobe. J. Math. \textbf{13} (1996), 123-132.

\bibitem{W} M. X. Wang;
\emph{Blow-up rate estimates for semilinear parabolic systems},
J. Differential equations. \textbf{170} (2001), 317-324.

\bibitem{Y} Y. F. Yin;
\emph{On nonlinear parabolic equations with nonlocal boundary
condition}, J. Math. Anal. Appl. \textbf{185} (1994), 54-60.

\bibitem{ZK} S. Zheng, L. Kong;
\emph{Roles of weight functions in a nonlinear nonlocal
parabolic system}, Nonlinear Anal. \textbf{68} (2008) 2406-2416.

\end{thebibliography}

\end{document}