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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 206, 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 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/206\hfil Simultaneous and non-simultaneous blow-up]
{Simultaneous and non-simultaneous blow-up and uniform blow-up
profiles for reaction-diffusion system}

\author[Z. Ling, Z. Wang \hfil EJDE-2012/206\hfilneg]
{Zhengqiu Ling, Zejia Wang}  % in alphabetical order

\address{Zhengqiu Ling \newline
Institute of Mathematics and Information Science,
 Yulin Normal University,  Yulin 537000,  China}
\email{lingzq00@tom.com}

\address{Zejia Wang \newline
 College of Mathematics and Information Science,
Jiangxi Normal University, Nanchang 330022, China}
\email{wangzj1979@gmail.com}

\thanks{Submitted August 6, 2012. Published November 24, 2012.}
\subjclass[2000]{35B33, 35B40, 35K55, 35K57}
\keywords{Simultaneous and non-simultaneous blow-up; \hfill\break\indent
 uniform blow-up profile; reaction-diffusion system; nonlocal sources}

\begin{abstract}
 This article concerns the blow-up solutions of a reaction-diffusion
 system with nonlocal sources, subject to the homogeneous Dirichlet
 boundary conditions. The criteria used to identify simultaneous and
 non-simultaneous blow-up of solutions by using the parameters $p$
 and $q$ in the model are proposed. Also,
 the uniform blow-up profiles in the interior domain are established.
\end{abstract}

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

\section{Introduction and description of results}

 In this article, we investigate the following reaction-diffusion system 
with nonlocal sources
 \begin{gather}  \label{1-1}
  u_t  = \Delta u +  \| u v \| ^p_\alpha  , \quad (x, t) \in \Omega\times (0, T), \\
 \label{1-2}
  v_t  = \Delta v \ +  \| u v \| ^q_\beta   , \quad (x, t) \in \Omega\times (0, T)\\
\label{1-3}
 u(x,0)=u_0(x), \quad v(x,0)=v_0(x), \quad x \in \Omega,\\
\label{1-4}  u(x,t)=0, \quad v(x,t)=0, \quad (x , t) \in \partial \Omega \times (0, T),
\end{gather}
where $ \Omega = B_R = \{ |x| < R \} \subset \mathbb{R}^N$ $(N\geq 1)$, 
$\alpha, \beta \geq 1$, $ p, q > 0$, and the continuous functions 
$u_0(x), v_0(x)$ are nonnegative, nontrivial, radially symmetric,
decreasing with $|x|$, and vanish on $\partial B_R$, where
 $\| \cdot \|^\alpha_\alpha = \int _\Omega |\cdot|^\alpha \,\mathrm{d}x$.


Nonlinear parabolic systems \eqref{1-1}-\eqref{1-4} can be used to describe some 
reaction diffusion phenomena, Such as heat propagations in a two-component 
combustible mixture \cite{1}, chemical reactions \cite{2}, interaction of two
biological groups without self-limiting \cite{3}, etc., where $u$ and $v$ represent 
the temperatures of two different
materials during a propagation, the thicknesses of two kinds of chemical reactants, 
the densities of two biological groups during a migration, etc. Using the methods 
of \cite{4,5,6} we know that \eqref{1-1}-\eqref{1-4} has a
local nonnegative classical solution.  Moreover, if $p, q \geq 1$,
 then the uniqueness holds.

In recent years, many results on blow-up solutions have been obtained for the 
nonlinear parabolic system. We will  recall several results in the following. 
As for the other related works on the global existence and
blow-up of solutions of the nonlinear parabolic system, they can be found 
in \cite{7,8,9,10} and references therein.

Li, Huang and Xie in \cite{11} and Deng, Li and Xie in \cite{12} considered 
the following two systems, respectively,
\[
 u_t = \Delta u + \int_\Omega u^m (x,t) v^n (x,t) \,\mathrm{d}x, \quad
 v_t = \Delta v + \int_\Omega u^p (x,t) v^q
(x,t) \,\mathrm{d}x, \]
with $x \in\Omega$, $t>0$; and
\[
u_t = \Delta u^m + a \|v\|^p_\alpha , \quad v_t = \Delta v^n + b \|u\|^q_\beta,
 \quad (x,t)\in \Omega \times (0,T).
\]
The authors showed some results on the global solutions, the blow-up solutions 
and the blow-up profiles. In 2002, Zheng, Zhao and Chen in \cite{13} studied 
the problem
\begin{equation}
u_t = \Delta u + f_1(u, v), \quad v_t = \Delta v + f_2(u, v), \quad (x,t)\in \Omega \times (0,T)
\end{equation}
with homogeneous Dirichlet boundary conditions, where
$$
f_1(u,v)=e^{m u(x,t) + p v(x,t)}, \quad f_2(u,v)=e^{q u(x,t) +  v(x,t)}.
$$
The simultaneous blow-up rates are obtained for radially symmetric blow-up 
solutions in the exponent region $\{ 0\leq m<q, 0\leq n<p\}$.

Later, Zhao and Zheng in \cite{14}, Li and Wang in \cite{15} studied the 
localized problem (1.5) with the more general
$\Omega \subset\mathbb{R}^N$ and
$$
f_1(u,v)=e^{m u(x_0, t) + p v(x_0, t)}, \quad
f_2(u,v)=e^{q u(x_0, t) + n v(x_0,t)}, \quad x_0\in\Omega.
$$
The critical blow-up exponents were discussed. Uniform blow-up profiles for 
simultaneous blow-up solutions were
proved in the exponent region $\{0\leq m \leq q, 0\leq n \leq p\}$.

Our present work is motivated by the above mentioned papers, 
the main purpose of this paper is to identify the
simultaneous and non-simultaneous blow-up of the solutions and establish 
the uniform blow-up profiles for the
system \eqref{1-1}--\eqref{1-4}.

For convenience, we introduce a pair of parameters $\sigma$ and $\theta$, 
the solution of
\begin{equation} \label{1-6}
\begin{pmatrix}
      p-1 & p      \\
      q  & q-1
\end{pmatrix}
\begin{pmatrix} \sigma \\ \theta \end{pmatrix} 
 = \begin{pmatrix} 1 \\ 1 \end{pmatrix},
\end{equation}
namely,
\begin{equation} \label{1-7} 
\sigma =\frac{p-(q-1)}{p+q-1}, \quad \theta = \frac{q-(p-1)}{p+q-1}\,.
\end{equation}

This paper is organized as follows. In the next Section, we investigate 
the simultaneous and non-simultaneous
blow-up of the solutions for the system \eqref{1-1}--\eqref{1-4}, 
and give the blow-up criteria. In Section 3, we deal with
the blow-up rates of the solutions.

\section{Simultaneous and non-simultaneous blow-up}

In this section, we discuss the simultaneous and non-simultaneous blow-up
 phenomena for the system \eqref{1-1}--\eqref{1-4},
and propose a complete and optimal classification to identify the simultaneous 
and non-simultaneous blow-up solutions.

For  problem \eqref{1-1}-\eqref{1-4}, because of  the nonlinear sources,
 there exist solution $(u, v)$ that
blow up in finite time, $T$, if and only if the exponents $p, q$ verify
 any of conditions, $p>1, q>1$ or $p q>(q-1)(p-1)$. In particular, 
the component $u ({\rm or }\ v) $ can blow up for the large initial data if
$p>q-1 ( {\rm or}\ q>p-1$), see [9, 12]. So there may be non-simultaneous 
blow-up, that is to say that one component blows up while the other remains bounded.
 On the other hand, the simultaneous blow-up means that
$$
\limsup_{t\to T}\|u(\cdot,t)\|_\infty = \limsup_{t\to T}\|v(\cdot,t)\|_\infty 
= + \infty.
$$
Assume the initial data $u_0(x), v_0(x)$ satisfy
\begin{gather}\label{2-1}
\Delta u_0(x) + \|u_0 v_0 \|^p_\alpha - \varepsilon \varphi(x) u_0^p(0) v_0^p(0) 
\geq 0, \quad x \in B_R,\\
\label{2-2}  \Delta v_0(x) + \|u_0 v_0 \|^q_\beta  
- \varepsilon \varphi(x)  u_0^q(0) v_0^q(0) \geq 0, \quad x \in B_R
\end{gather}
for some a constant $\varepsilon \in (0, 1)$, where $\varphi(x)$ is 
the first eigenfunction of
$$
-\Delta \varphi = \lambda \varphi ,\;  x\in B_R; \quad 
\varphi =0, \; x\in \partial B_R,
$$
normalized by $ \|\varphi\|_\infty =1, \ \varphi>0$ in $B_R$.
 In addition, by using the methods in \cite{16}, it is
easy to check that $u_t, v_t \geq 0$ for $(x,t) \in B_R \times (0,T)$ 
by the comparison principle.

Our results about the simultaneous and non-simultaneous blow-up criteria
 are as follows.

\begin{theorem}\label{thm2-1}
 If $p+q>1$, then there exists initial data such that the non-simultaneous 
blow-up occurs in \eqref{1-1}--\eqref{1-4} if and only if $ \sigma < 0$
 $($or $\theta < 0 )$ $($ for $v ($or $u)$ blowing up alone,
respectively$)$.
\end{theorem}

\begin{theorem}\label{thm2-2} 
If $p+q>1$, then any blow-up in \eqref{1-1}--\eqref{1-4} is non-simultaneous 
if and only if $\sigma\geq
0$ with $\theta < 0$ $($ for $u$ blowing up alone $)$, or
 $\theta \geq 0$ with $\sigma < 0$ $($ for $v$ blowing up alone$)$.
\end{theorem}

\begin{corollary} \label{coro2-1} 
If $p+q>1$, then any blow-up in \eqref{1-1}--\eqref{1-4} is simultaneous 
if and only if $\sigma \geq 0$ and $\theta \geq 0$.
\end{corollary}

Similar to the study in\cite{11}, it is seen that

\begin{corollary} \label{coro2-2} 
All solutions are global in \eqref{1-1}--\eqref{1-4} if and only
 if $\sigma<0$ and $\theta <0 ($i.e., $p+q<1)$.
\end{corollary}

In summary, the complete and optimal classification for simultaneous 
and non-simultaneous blow-up solutions of the problem
\eqref{1-1}-\eqref{1-4} can be described by Figure \ref{fig1}

\begin{figure}[ht]
\begin{center}
\begin{picture}(240,90)(-20,-10)
\put(-20,0){\vector(1,0){240}} 
\put(0,-10){\vector(0,1){90}} 
\put(42,-5){\line(5,3){120}}
\put(-5,35){\line(5,3){60}} 
\put(-25,-10){(0, 0)} 
\put(48,-10){$1$ } 
\put(210,-10){$p$} 
\put(-12, 38){ $1$}
\put(-10,75){$q$} 
\put(55,70){$ p=q-1$} 
\put(160,70){ $ q=p-1$ }
\put(-4.5,42){\line(5,-4){60}} 
\put(21,21){$p+q=1$}
\put(45,41){\shortstack{\shortstack{simultaneous \\blow-up}}} 
\put(95,14){\shortstack{$u$ blows up alone \\ (non-simultaneous blow-up)}} 
\put(1,62){\shortstack{$v$ blows up \\ alone}} 
\put(4,6){global}
\end{picture}
\end{center}
\caption{Regions of simultaneous and non-simultaneous blow-up}
\label{fig1}
\end{figure}

The key clues for the classification of simultaneous and non-simultaneous 
blow-up solutions are the signs of
$p-(q-1)$, $q-(p-1)$ and $p+q-1$. The conditions $p>q-1$ and $p+q>1$ 
imply that $u$ may blow up by itself but
cannot provide sufficient help to the blow-up of $v$ 
(with small $v_0$), while $q < p-1$ ensures that $v$ can
provide effective help to the blow-up of $u$, but $v$ remains bounded.

Before we give the proof of Theorem \ref{thm2-1}, we first introduce 
the following lemma.  Let $\phi(x,t)$ satisfy
$$
\phi_t = \Delta \phi, \; (x,t)\in B_R\times (0,T); \quad 
\phi =0, \; (x,t)\in \partial B_R \times (0,T)
$$
with
$$
\phi(x,0)=\varphi(x), \quad x \in B_R.
$$
\begin{lemma} \label{lem2-1}
Under  conditions \eqref{2-1} and $\eqref{2-2}$, the solution $(u,v)$ 
of \eqref{1-1}--\eqref{1-4} satisfies
\begin{gather} \label{2-3} 
 u_t(x,t) \geq \varepsilon \phi(x,t) u^p(0,t) v^p(0,t) , \quad
 (x,t)\in B_R\times [0,T),\\
\label{2-4} 
 v_t(x,t) \geq \varepsilon \phi(x,t) u^q(0,t) v^q(0,t), \quad
 (x,t)\in B_R\times [0,T).
\end{gather}
\end{lemma}


\begin{proof} 
Since that the proofs of the inequalities \eqref{2-3} and \eqref{2-4} are similar,
we prove only \eqref{2-3}. Let
$$
J(x,t)= u_t(x,t)-\varepsilon \phi(x,t)  u^p(0,t) v^p(0,t).
$$
It is easy to check that for $\varepsilon$ small enough since 
$u_t, v_t \geq 0$, we obtain
\begin{gather*}
 J_t - \Delta J = \big(\|uv\|^p_\alpha\big)_t 
- \varepsilon \phi \big( u^p(0,t)v^p(0,t)\big)_t \geq 0, \quad
(x,t)\in B_R \times (0,T), \\
 J(x,t)=0, \quad (x,t)\in \partial B_R \times (0,T), \\
 J(x,0)=\Delta u_0(x) + \|u_0 v_0 \|^p_\alpha 
- \varepsilon \varphi(x) u_0^p(0) v_0^p(0) \geq 0, \quad x \in B_R.
\end{gather*}
Consequently,  \eqref{2-3} is true by the comparison principle. 
\end{proof} 

\begin{proof}[Proof of Theorem \ref{thm2-1}]
Without loss of generality, we only prove that there exist suitable
initial data such that $u$ blows up while $v$ remains bounded if and 
only if $\theta < 0$.

Assume $\theta < 0$, namely, $p-1>q$ and $p>1$ by Figure \ref{fig1}
 and \eqref{1-7}. From \eqref{2-3}, we obtain that
\begin{equation} \label{2-5} 
u_t(0,t) \geq \varepsilon \phi (0,T) u^p(0,t) v_0^p(0), \quad t \in [0,T).
\end{equation}
Integrating the above inequality \eqref{2-5} from $t$ to $T$,
 we have the estimate for $u$ as follows
\begin{equation}\label{2-6} 
u(0,t) \leq \Big( \varepsilon (p-1) \phi(0,T) v_0^p(0)
\Big)^{-1/(p-1)}(T-t)^{-1/(p-1)},\quad t\in [0,T).
\end{equation}
At the same time, since the initial data $(u_0, v_0)$ is radially symmetric 
and non-increasing, therefore the
$(u, v)$ is also radial symmetrical and non-increasing; i.e.,
 $u_r(r,t), v_r(r,t)\leq 0$ for $r\in [0, R)$.
Thus, $u(x,t)$ and $v(x,t)$ always reach their maxima at $x=0$, which means that
$$
 \Delta u(0,t) \leq 0, \quad \Delta v(0,t)\leq 0.
$$
Hence, from \eqref{1-1} and \eqref{1-2}, we know that there exist 
constants $C_1, C_2>0$ such that
\begin{equation}  \label{2-7}
\begin{gathered}
 u_t (0, t) \leq \|uv\|_\alpha^p \leq C_1 u^p(0,t) v^p(0,t), \quad t\in [0,T) \\
 v_t (0, t) \leq \|uv\|_\beta^q \leq C_2 u^q(0,t) v^q(0,t), \quad t\in [0,T).
\end{gathered}
\end{equation}
Let
$$
\Gamma(x,y,t,s)= \frac{1}{[4 \pi (t-s)]^{N/2}}
\exp\big\{-\frac{|x-y|^2}{4(t-s)}\big\}
$$
be the fundamental solution of the heat equation.
 Suppose that $(\tilde{u}_0, \tilde{v}_0)$ is a pair of initial
data such that the solution of \eqref{1-1}--\eqref{1-4} blows up. 
Fix radially symmetrical $v_0 (\geq\tilde{v}_0 )$ in $B_R$
and take constant $M_1 > v_0(x)$. By the proof of \cite[Theorem 1.1]{17}, 
we know that if $u_0$ is large with $v_0$
fixed then $T$ becomes small. Therefore, let $u_0(\geq \tilde{u}_0)$ 
be large such that $T$ becomes small and satisfies
$$
M_1 \geq v_0(0) + \frac{p-1}{p-1-q} \big( \varepsilon (p-1) \phi(0,T)
v_0^p(0)\big)^{-\frac{q}{p-1}}T^{\frac{p-1-q}{p-1}}\|M_1\|^q_\beta,
$$
where $\| M_1 \|_\beta^q = ( \int_\Omega M_1^\beta \,\mathrm{d}x )^{q/\beta}$. 
Consider the following auxiliary problem
\begin{gather*}
 \bar{v}_t = \Delta \bar{v} + \big( \varepsilon (p-1) \phi(0,T)
v_0^p(0)\big)^{-\frac{q}{p-1}}(T-t)^{-\frac{q}{p-1}} \|M_1\|^q_\beta, 
\quad  (x,t)\in B_R \times (0,T),\\
 \bar{v}(x,t)=0, \quad (x,t)\in \partial B_R \times (0,T),\\
 \bar{v}(x,0) = v_0(x),\quad  x\in B_R.
\end{gather*}
Since $p-1>q$, we obtain by Green's identity that
$$
\bar{v} \leq v_0(0) + \frac{p-1}{p-1-q}
\Big(\varepsilon (p-1) \phi(0,T) v_0^p(0) \Big) ^{-\frac{q}{p-1}} T
^{\frac{p-1-q}{p-1}} \|M_1\|^q_\beta \leq M_1,
$$
and hence $\bar{v}$ satisfies
$$
\bar{v}_t \geq \Delta \bar{v} + \big( \varepsilon (p-1) \phi(0,T) v_0^p(0)
\big)^{-\frac{q}{p-1}}(T-t)^{-\frac{q}{p-1}} \|\bar{v}(x,t)\|^q_\beta.
$$
On the other hand, $v$ satisfies
$$
v_t \leq \Delta v + \big( \varepsilon (p-1) \phi(0,T) v_0^p(0) \big)^{-\frac{q}{p-1}}(T-t)^{-\frac{q}{p-1}}
\|v(x,t)\|^q_\beta.
$$
Therefore, by the comparison principle, we conclude $v \leq \bar{v}\leq M_1$.

Now assume that $u$ blows up while $v$ remains bounded. By \eqref{2-7} we have
\[
u_t(0,t) \leq C u^p(0,t), \quad \text{for } t \in [0,T).
\]
 This implies $p>1$ and the estimate for $u$ that
$$
u(0,t) \geq \big( C (p-1) \big)^{-1/(p-1)}(T-t)^{-1/(p-1)}.
$$
Therefore, by using \eqref{2-4}, we have
$$
v_t(0,t) \geq \varepsilon \phi(0,T)\big(C(p-1)\big)^{-\frac{q}{p-1}} v_0^q(0) (T-t)^{-\frac{q}{p-1}}.
$$
By integrating, we obtain that
\begin{equation} \label{2-9} 
v(0,t) \geq v_0(0) + \varepsilon \phi(0,T)\big(C(p-1)\big)^{-\frac{q}{p-1}} 
v_0^q(0) \int_0^t (T-s)^{-\frac{q}{p-1}} \,\mathrm{d}s.
\end{equation}
The boundedness of $v$ requires $p-1>q$ from \eqref{2-9}, that is $\theta<0$.
 Thus, the proof is complete.
\end{proof} 


\begin{proof}[Proof of Theorem \ref{thm2-2}]
 We only treat the case of $u$ blowing up and $v$ remains bounded.

Assume $\sigma \geq 0$ with $\theta < 0$; that is $p \geq q-1, q < p-1$ 
and $p>1$ by Figure \ref{fig1} and \eqref{1-7}. From
\eqref{2-3} and \eqref{2-7}, we have
\begin{equation}  \label{2-10} 
v^{p-q}(0,t) v_t(0,t) \leq \frac{C_2}{\varepsilon \phi(0,T)}u^{q-p}(0,t) u_t(0,t),
\quad t\in[0,T).
\end{equation}
By Theorem \ref{thm2-1}, it is impossible for $v$ blowing up alone under $\sigma \geq 0$ 
with $\theta<0$. Then we show
that $v$ is bounded. In fact, by integrating the inequality \eqref{2-10} 
from $0$ to $t$, we have
$$
 v^{p-q+1}(0,t) \leq C - C u^{-(p-q-1)}(0,t)
$$
for some a  $C>0$. Therefore, we can get the boundedness of $v(0,t)$.

Now, assume that any blow-up must be the case for $u$ blowing up alone. 
This requires $\theta <0$ by Theorem \ref{thm2-1}. Again by 
Theorem \ref{thm2-1}, 
if in addition $\sigma<0$, there exists the initial data such that $v$ blows 
up alone.
Therefore, it has to be satisfied that $\sigma \geq 0$. 
Then, the proof is complete. 
\end{proof}

\section{Uniform Blow-up Profiles}

In this section, we study the uniform blow-up profiles for 
system \eqref{1-1}--\eqref{1-4}. At first, the following
result of Souplet for a single diffusion equation with nonlocal nonlinear 
sources \cite[Theorem 4.1]{18} will play a basic role in our discussion.

\begin{lemma}\label{lem3-1} 
Let $u\in C^{2,1}(\bar{\Omega}\times (0,T^*))$ be a solution of the problem
\begin{gather*}
 u_t = \Delta u + g(t), \quad (x,t)\in \Omega \times (0,T^*),\\
 u(x,t)=0, \quad (x,t)\in \partial\Omega \times (0,T^*),\\
 u(x,0)=u_0(x), \quad x\in\Omega,
\end{gather*}
where $g(t)$ is nonnegative and may depend on the solution $u$. Then
\begin{equation} \label{3-1} 
\lim_{t \to T^*}\|u(\cdot,t)\|_\infty = +\infty
\end{equation}
if and only if $\int_0^t g(s) \,\mathrm{d}s = +\infty$. 
Furthermore, if $\eqref{3-1}$ is fulfilled, then
$$
\lim_{t \to T^*}\frac{u(x,t)}{G(t)} 
= \lim_{t \to T^*}\frac{\|u(\cdot,t)\|_\infty}{G(t)} = 1
$$
uniformly on compact subsets of $\Omega$, where $G(t)=\int_0^t g(s)\,\mathrm{d}s$.
\end{lemma}

For convenience, we denote
$$
f(t)=\| u v \|^p_\alpha, \quad  g(t)=\|u v \|^q_\beta, \quad  
F(t)=\int_0^t f(s) \,\mathrm{d}s, \quad
G(t)=\int_0^t g(s) \,\mathrm{d}s.
$$
According to the Lemma \ref{lem3-1}, we have the following result.

\begin{lemma} \label{lem3-2}
 Assume $u, v \in C^{2,1}(\bar{\Omega}\times [0,T))$ are the solutions 
of \eqref{1-1}--\eqref{1-4}. If $u$ and
$v$ blow up simultaneously in the finite time $T^*$, then we have
$$
\lim_{t \to T^*} \frac{u(x,t)}{F(t)}= 1, \quad 
\lim_{t \to T^*} \frac{v(x, t)}{G(t)}= 1
$$
uniformly on compact subsets of $\Omega$, and
$$
\lim_{t \to T^*} F(t) = \lim_{t \to T^*} G(t) = \infty.
$$
\end{lemma}

We remark that if we assume that only $u$ (or $v$) blows up in finite 
time $T^*$, then the above conclusions about
$u$ ( or $v$) and $F$ (or $G$) are also valid.

Throughout this section the notation $f(t)\sim g(t)$ is used to describe 
such functions $f(t)$ and $g(t)$ satisfying $f(t)/g(t) \to 1$ as $t \to T^*$.
When $u$ and $v$ blow up simultaneously, we have the following results 
about the uniform blow-up profiles for $u$ and $v$.

\begin{theorem} \label{thm3-1} 
Let $(u,v)$ be a solution of \eqref{1-1}--\eqref{1-4} with simultaneous 
blow-up time $T^*$. Then the
 following limits hold uniformly on any compact subset of $\Omega$:

$(1)$ If $\sigma >0$ and $\theta>0$, then 
\begin{gather}\label{3-2} 
\lim_{t\to T^*} u(x,t) (T^*-t)^\sigma = \Big(
\frac{|\Omega|^{p/\alpha}}{\sigma}(|\Omega|^{\frac{q}{\beta}-\frac{p}{\alpha}}\frac{\sigma}
{\theta})^{p/(p+1-q)}\Big)^{-\sigma},
\\
\label{3-3}
 \lim_{t\to T^*} v(x,t) (T^*-t)^\theta = \Big(
\frac{|\Omega|^{q/\beta}}{\theta}(|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}
\frac{\theta}{\sigma})^{q/(q+1-p)}\Big)^{-\theta}.
\end{gather}

$(2)$ If $\sigma = 0$, then 
\begin{gather}\label{3-4} 
 \lim_{t\to T^*} u^2(x,t)|\ln (T^* -t)|^{-1} = \frac{2}{p}|\Omega|^{\frac{p}{\alpha} -
\frac{q}{\beta}},
\\
\label{3-5} \lim_{t\to T^*} v^p(x,t)\big(\ln
v(x,t)\big)^{\frac{q}{2}}(T^*-t)=\frac{1}{p}|\Omega|^{-q/\beta}\big(2
|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}\big)^{-q/2}.
\end{gather}

$(3)$ If $\theta=0$, then we have
\begin{gather} \label{3-6} 
\lim_{t\to T^*} u^q(x,t)\big(\ln u(x,t)\big)^{\frac{p}{2}}(T^*-t) =
\frac{1}{q}|\Omega|^{-p/\alpha}\big( 2 |\Omega| ^{\frac{q}{\beta} - \frac{p}{\alpha}}\big)^{-p/2},
\\
\label{3-7} 
\lim_{t\to T^*} v^2(x,t)|\ln (T^*-t)|^{-1}=
\frac{2}{q}|\Omega|^{\frac{q}{\beta}-\frac{p}{\alpha}}.
\end{gather}
\end{theorem}

\begin{proof}
 From Lemma \ref{lem3-2}, we know that $u(x,t) \sim F(t)$ and $v(x,t) \sim G(t)$, then
\begin{gather*}
\lim_{t \to T^* } \frac{u^\alpha(x,t)}{F^\alpha(t)} = \lim_{t \to T^* }
\frac{v^\alpha(x,t)}{G^\alpha(t)} = 1,
\\
\lim_{t \to T^* } \frac{u^\beta(x,t)}{F^\beta(t)} = \lim_{t \to T^* }
\frac{v^\beta(x,t)}{G^\beta(t)} = 1.
\end{gather*}
By the Lebesgue dominated convergence theorem, we find that
\begin{gather} \label{3-8}
  F' (t) = f(t) = \|uv\|^p_\alpha \sim |\Omega|^{p/\alpha} F^p(t) G^p(t),
\\
\label{3-9}  G' (t) = g(t) = \|uv\|^q_\beta  \sim |\Omega|^{q/\beta}  F^q(t) G^q(t).
\end{gather}
Hence,
\begin{equation}
\label{3-10} F^{q-p} \,\mathrm{d} F \sim |\Omega|^{\frac{p}{\alpha}
-\frac{q}{\beta}} G^{p-q} \,\mathrm{d}G.
\end{equation}

(1) Note that the conditions $\sigma>0$ and $\theta>0$ imply that $p+1>q, q+1>p$ 
since $p+q>1$. Integrating \eqref{3-10} from $0$ to $t$,  we obtain
\begin{equation}
\label{3-11} F^{q+1-p}(t) \sim |\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}} \frac{q+1-p}{p+1-q} G^{p+1-q}(t) =
|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}} \frac{\theta}{\sigma}G^{p+1-q}(t).
\end{equation}
Combining \eqref{3-9} and \eqref{3-11}, we can obtain
\begin{equation}
\label{3-12} G' (t) \sim |\Omega|^{q/\beta} \big( |\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}
\frac{\theta}{\sigma} \big) ^{\frac{q}{q+1-p}} G^{\frac{2q}{q+1-p}}(t).
\end{equation}
Since
$$
1 - \frac{2q}{q+1-p} = -\frac{p+q-1}{q+1-p} = - \frac{1}{\theta}<0
$$
and $\lim_{t \to T^*} G(t) = \infty$, by integrating \eqref{3-12}, we obtain
\begin{equation}
\label{3-13} G(t) \sim \Big( \frac{|\Omega|^{q/\beta}}{\theta} \big( |\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}
\frac{\theta}{\sigma} \big) ^{\frac{q}{q+1-p}}\Big)^{-\theta} (T^* -t) ^{-\theta}.
\end{equation}
From \eqref{3-13} and Lemma \ref{lem3-2}, we have
$$
\lim_{t \to t^*} v(x,t)(T^*-t)^\theta
 = \Big( \frac{|\Omega|^{q/\beta}}{\theta} \big(
|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}} \frac{\theta}{\sigma} \big) ^{\frac{q}{q+1-p}}\Big)^{-\theta},
$$
which holds uniformly on the compact subsets of $\Omega$.

Combining \eqref{3-8} and \eqref{3-11}, and applying the similar proofs of
 $F$ and $u$, we obtain that
$$
\lim_{t\to T^*} u(x,t) (T^*-t)^\sigma = \Big(
\frac{|\Omega|^{p/\alpha}}{\sigma}\big(|\Omega|^{\frac{q}{\beta}-\frac{p}{\alpha}}\frac{\sigma}
{\theta}\big)^{\frac{p}{p+1-q}}\Big)^{-\sigma}
$$
holds uniformly on the compact subsets of $\Omega$.

(2) When $\sigma=0$, or $p+1=q$, noticing \eqref{3-9} and \eqref{3-10}, we see that
\begin{equation} \label{3-14}
 G' (t) \sim |\Omega|^{q/\beta}\big(2 |\Omega|^{\frac{p}{\alpha}
-\frac{q}{\beta}}\big)^{q/2} G^q(t) \big( \ln G(t)\big) ^{q/2}.
\end{equation}
Note that $\lim_{t \to T^*} G(t) = \infty$, integrating \eqref{3-14}
 from $t(>0)$ to $T^*$ asserts
\begin{equation}
\label{3-15} \int_{G(t)}^\infty \frac{1}{s^q (\ln s)^{q/2}} \,\mathrm{d}s \sim |\Omega|^{q/\beta}\big(2
|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}\big)^{q/2} (T^* -t).
\end{equation}
Furthermore,
$$
\lim_{t \to T^*} \frac{\int_{G(t)}^\infty s^{-q} (\ln s)^{-q/2}\,\mathrm{d}s}{G^{1-q}(t)\big(\ln
G(t)\big)^{-q/2}} = \lim_{G \to \infty} \frac{\int_{G}^\infty s^{-q} (\ln
s)^{-q/2}\,\mathrm{d}s}{G^{1-q}\big(\ln G\big)^{-q/2}} = \frac{1}{q-1}=\frac{1}{p}.
$$
That is to say that
\begin{equation}\label{3-16} 
p \int_{G(t)}^\infty s^{-q} (\ln s)^{-q/2} 
\,\mathrm{d}s \sim G^{1-q}(t) ( \ln G(t) ) ^{-q/2} =
G^{-p}(t) ( \ln G(t) ) ^{-q/2}.
\end{equation}
By \eqref{3-15} and \eqref{3-16}, it indicates
\begin{equation} \label{3-17}
 G^{-p}(t)( \ln G(t) ) ^{-q/2}  \sim p |\Omega|^{q/\beta} \big(2
|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}\big)^{q/2} (T^*-t).
\end{equation}
Since $\lim_{t \to T^*} v(x,t)=\infty$ uniformly on the compact subset
 of $\Omega$ and $\lim_{t \to T^*} G(t) = \infty$, we may claim that the 
following equivalent is valid uniformly on the compact subset of
$\Omega$,
$$
v(x,t) \sim G(t) \; \Rightarrow\; \ln v(x,t) \sim \ln G(t).
$$
And thus by \eqref{3-17}, we reach the conclusion
$$
v^{-p}(x,t)( \ln v(x,t) ) ^{-q/2}  \sim p |\Omega|^{q/\beta} \big(2
|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}\big)^{q/2} (T^*-t).
$$
Then uniformly on the compact subsets of $\Omega$, it yields
$$
\lim_{t \to T^*} v^p(x,t) (\ln v(x,t))^{q/2}(T^*-t) = \frac{1}{p}|\Omega|^{-q/\beta}\big( 2
|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}\big)^{-q/2}.
$$
Since
 $$
  \ln G(t) \sim \frac{1}{2} |\Omega|^{\frac{q}{\beta}-\frac{p}{\alpha}} F^2 (t),
 $$
it follows from \eqref{3-8} and \eqref{3-17} that
\begin{equation}
\label{3-18}
 F' (t) F^{-p}(t) \sim |\Omega|^{p/\alpha} G^p(t) \sim
\frac{F^{-q}(t)}{p(T^*-t)}|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}.
\end{equation}
In view of \eqref{3-18}, we have
 $$
 \frac{1}{2} F^2 (t) \sim \frac{1}{p}|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}
|\ln(T^*-t)|.
$$
Therefore, by Lemma \ref{lem3-2}, we obtain
$$
u^2(x,t) \sim \frac{2}{p}|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}} |\ln(T^*-t)|;
$$
that is to say
$$
\lim_{t \to T^*} u^2(x,t) |\ln (T^* -t)|^{-1} = \frac{2}{p}|\Omega|^{\frac{p}{\alpha}-\frac{q}{\beta}}
$$
holds uniformly on the compact subsets of $\Omega$.

 (3) When $\theta =0$, the proof is similar to that of the case (2).
Then, the proof is completed. 
\end{proof}

\subsection*{Acknowledgements}
This work was supported by the NNSF of China. 
The authors want to thank the anonymous referees for their helpful suggestions.

\begin{thebibliography}{00}

\bibitem{8} H. W. Chen; 
\emph{Global existence and blow-up for a nonlinear reaction-diffusion system}, 
 J. Math. Anal. Appl., 212 (1997), 481-492.

\bibitem{12} W. B. Deng, Y. X. Li, C. H. Xie; 
\emph{Blow-up and global existence for a nonlocal degenerate parabolic system},
J. Math. Anal. Appl., 277 (2003), 199-217.

\bibitem{1} E. Escobedo, M. A. Herrero; 
\emph{Boundedness and blow-up for a semilinear reaction-diffusion system},  J.
Diff. Equations, 89(1) (1991), 176-202.

\bibitem{6} M. Escobedo, M.A. Herrero; 
\emph{A semilinear parabolic system in a bounded domain},  Ann. Mate. Pura Appl.,
 CLXV(IV) (1993), 315-336.

\bibitem{9} M. Escobedo, H. A. Levine; 
\emph{Critical blow-up and global existence numbers for a weakly coupled system of
reaction-diffusion equations},  Arch. Rational Mech. Anal., 129 (1995), 47-100.

\bibitem{2} P. Glansdorff, I. Prigogine; 
\emph{Thermodynamic Theory of Structure, Stability and Fluctuation},
Wiley-interscience, London, 1971.

\bibitem{4} O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva;
 \emph{Linear and Quasi-linear Equations of Parabolic Type},
Amer. Math. Soc., Rhode Island, 1968.

\bibitem{11} F. C. Li, S. X. Huang, C. H. Xie; 
\emph{Global existence and blow-up of solutions to a nonlocal reaction-diffusion
system},  Discrete Contin Dyn. Syst., 9(6) (2003), 1519-1532.

\bibitem{15} H. L. Li, M. X. Wang;
 \emph{Blow-up properties for parabolic systems with localized nonlinear sources},  
Appl. Math. Lett., 17 (2004), 771-778.

\bibitem{3} H. Meinhardt; 
\emph{Models of Biological Pattern Formation}, Academic Press, London, 1982.

\bibitem{17} F. Quir\'os, J. D. Rossi; 
\emph{Non-simultaneous blow-up in a semilinear parabolic system},
Z. angew. Math. Phys., 52 (2001), 342-346.

\bibitem{5} P. Souplet; 
\emph{Blow up in nonlocal reaction-diffusion equations}, SIMA J. Math. Anal.,
29(6) (1998), 1301-1334.

\bibitem{18} P. Souplet; 
\emph{Uniform Blow-up profiles and boundary behavior for diffusion equations 
with nonlocal nonlinear source},  J. Diff. Equations, 153 (1999), 374-406.

\bibitem{10} L. W. Wang; 
\emph{The blow-up for weakly coupled reaction-diffusion system}, 
 Proc. Amer. Math. Soc., 129 (2000), 89-95.

\bibitem{7} M. X. Wang;
 \emph{Global existence and finite time blow-up for a reaction-diffusion system},  
Z. Angew. Math. Phys., 51 (2000), 160-167.

\bibitem{16} M. X. Wang; 
\emph{Blow-up rate estimates for semilinear parabolic systems}, 
J. Diff. Equations, 170(2001)317-324.

\bibitem{14} L. Z. Zhao, S. N. Zheng; 
\emph{Critical exponents and asymptotic estimates of solutions to parabolic 
systems with localized nonlinear sources}, J. Math. Anal. Appl., 292 (2004), 
621-635.

\bibitem{13} S. N. Zheng, L. Z. Zhao, F. Chen;
 \emph{Blow-up rates in a parabolic system of ignition model},  Nonlinear
Anal. 51 (2002), 663-672.

\end{thebibliography}

\end{document}