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

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations}, 
Vol. 2006(2006), No. 94, pp. 1--8.\newline 
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/94\hfil A blow up condition]
{A blow up condition for a nonautonomous semilinear system}

\author[A. P\'erez\hfil EJDE-2006/94\hfilneg]
{Aroldo P\'erez-P\'erez}

\address{Aroldo P\'erez-P\'erez\hfill\break
Divisi\'on Acad\'emica de Ciencias B\'asicas, 
Universidad Ju\'arez Aut\'onoma de Tabasco,
 Km. 1 Carretera Cunduac\'an-Jalpa de M\'endez, C.P. 86690 A.P. 24,
Cunduac\'an, Tabasco, M\'exico}
\email{aroldo.perez@dacb.ujat.mx}

\date{}
\thanks{Submitted July 14, 2006. Published August 18, 2006.}
\subjclass[2000]{35B40, 35K45, 35K55, 35K57}
\keywords{Finite time blow up; mild solution; weakly coupled system;
\hfill\break\indent nonautonomous initial value problem;
fractal diffusion}

\begin{abstract}
 We give a sufficient condition for finite time blow up of the
 nonnegative mild solution to a nonautonomous weakly coupled system
 with fractal diffusion having a time dependent factor  which is
 continuous and nonnegative.
\end{abstract}

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

\section{Introduction}

This paper deals with the blow up of nonnegative solutions of the
nonautonomous initial value problem for a weakly coupled system
with a fractal diffusion
\begin{equation} \label{general}
\begin{aligned}
\frac{\partial u(t,x)}{\partial
t}&=k(t)\Delta_{\alpha}u(t,x)+v^{\beta_{1}}(t,x),\quad t>0,\quad
x\in \mathbb{R}^{d}  \\
\frac{\partial v(t,x)}{\partial
t}&=k(t)\Delta_{\alpha}v(t,x)+u^{\beta_{2}}(t,x),\quad t>0,\quad
x\in \mathbb{R}^{d}\\
 u(0,x)&=\varphi_{1}(x),\quad x\in \mathbb{R}^{d}\\
v(0,x)&=\varphi_{2}(x),\quad x\in \mathbb{R}^{d},
\end{aligned}
\end{equation}
where $\Delta _{\alpha }:=-(-\Delta )^{\alpha /2}$,
$0<\alpha\le2$ denotes the $\alpha $-Laplacian, 
$\beta_{1},\text{ }\beta_{2} >1$ are constants,
$0\leq\varphi_{1}, \text{ }\varphi_{2}\in B(\mathbb{R}^{d})$
(where $B(\mathbb{R}^{d})$ is the space of bounded measurable
functions on $\mathbb{R}^{d})$ do not vanish identically,
$k:[0,\infty )\to [0,\infty )$ is continuous and satisfies
\begin{equation}
\varepsilon _{1}t^{\rho }\leq
\int_{0}^{t}k( r) dr\leq \varepsilon
_{2}t^{\rho },\quad \varepsilon _{1}, \;\varepsilon
_{2},\; \rho >0, \label{condk}
\end{equation}
for all $t$ large enough.

 The associated integral system to \eqref{general} is given by
 \begin{gather}
u(t,x)=U(t,0)\varphi_{1}(x)+\int_{0}^{t}U(t,r)v^{\beta_{1}}(r,x)dr,\quad
t>0,\text{ }x\in\mathbb{R}^{d},\label{mildsol1}\\
v(t,x)=U(t,0)\varphi_{2}(x)+\int_{0}^{t}U(t,r)u^{\beta_{2}}(r,x)dr,\quad
t>0,\text{ }x\in\mathbb{R}^{d},\label{mildsol2}
\end{gather}
where $\{U(t,s)\}_{t\geq s\geq 0}$ is the evolution family on
$B(\mathbb{R}^{d})$ that solves the homogeneous Cauchy problem for
the family of generators $\{k(t)\Delta_{\alpha}\}_{t\geq0}$.
Clearly
\begin{equation*}
U(t,s)=S(K(t,s)),\quad t\geq s\geq0,
\end{equation*}
where $\{S(t)\}_{t\geq0}$ is the semigroup with infinitesimal
generator $\Delta_{\alpha}$, and
$K(t,s)=\int_{s}^{t}k(r)dr$, $t\geq s\geq0$.

A solution of (\ref{mildsol1})-(\ref{mildsol2}) is called a mild
solution of \eqref{general}. If there exist a solution $(u,v)$ of
\eqref{general} in $[0,\infty)\times\mathbb{R}^{d}$ such that
$\| u( t,\cdot)\|_{\infty}+\| v(t,\cdot)\|_{\infty}<\infty$ for 
any $t\geq 0$, we say
that $(u,v)$ is a global solution, and when there exist a number
$T_{\varphi_{1},\varphi_{2}}<\infty$ such that \eqref{general} has
a bounded solution $(u,v)$ in $[0,T]\times \mathbb{R}^{d}$ for all
$T<T_{\varphi_{1},\varphi_{2}}$ with 
$\lim_{t\uparrow T_{\varphi_{1},\varphi_{2}}}\| u( t,\cdot)
\|_{\infty}=\infty$ or 
$\lim_{t\uparrow T_{\varphi_{1},\varphi_{2}}}\| v( t,\cdot)
\|_{\infty}=\infty$ we say that $(u,v)$ blows up in finite
time.

The finite time blow up of \eqref{general} for ${\alpha}=2$ and
$k\equiv1$ was initially considered by Escobedo and Herrero
\cite{E-H}. They proved that when ${\beta_{1}}{\beta_{2}}>1$ and
$(\gamma+1)/({\beta_{1}}{\beta_{2}}-1)\geq d/2$ with
$\gamma=\max\{\beta_{1},\beta_{2}\}$, any nontrivial positive
solution to \eqref{general} blows up in finite time. Related
results and more general cases for the Laplacian can be found for
instance in \cite{A-H-V,De,E-L,F-L-U,Ko,M-H,S-T,Wa,Zh}. The case
for fractional powers of the Laplacian when $k\equiv1$ for
equations with different diffusion operators was considered in
\cite{Gu-Ki,K-Q}; see also \cite{B-L-W,L-W} for a probabilistic
approach. Sugitani \cite{Su} has considered a scalar version of
\eqref{general} with $k\equiv1$ when the nonlinear term is given
by an increasing nonnegative continuous and convex function
$F(u)$, defined on $[0,\infty)$, and Guedda and Kirane \cite{G-K}
have considered a scalar version of \eqref{general} with
$k\equiv1$ when the nonlinear term is $h(t)u^{\beta}$,
$\beta>1$ with $h$ being a nonnegative continuous function on
$[0,\infty)$ satisfying $c_{0}t^{\sigma}\leq h(t)\leq
c_{1}t^{\sigma}$ for sufficiently large $t$, where $c_{0},
c_{1}>0$ and $\sigma>-1$ are constants. They proved that in this
scalar case, solutions blow up in finite time if
$0<d(\beta-1)/\alpha\leq1+\sigma$ for any nontrivial nonnegative
and continuous initial function on $\mathbb{R}^{d}$. Here we prove
that if $k$ satisfies (\ref{condk}) and
$0<d\rho(\beta_{i}-1)/\alpha<1$, $i=1,2$, then any
nontrivial positive solution of \eqref{general} blows up in finite
time. Here solutions will be understood in the mild sense, that
is, that solve (\ref{mildsol1})-(\ref{mildsol2}).

\section{Blow up condition}

Let $(u(\cdot,\cdot),v(\cdot,\cdot))$ be a nonnegative solution of
\eqref{general} and define
\begin{equation*}
u(t)=\int_{\mathbb{R}^{d}}p(K(t,0),x)u(t,x)dx,\quad
v(t)=\int_{\mathbb{R}^{d}}p(K(t,0),x)v(t,x)dx,\quad t>0,
\end{equation*}
where $p(t,x)$, $t>0$, $x\in\mathbb{R}^{d}$ denotes the
density of the semigroup $S(t)$, $t\geq0$.

\begin{lemma}  \label{Lemma1} 
For any $s,t>0,$ and  $x,y\in \mathbb{R}^{d}$, 
$p( t,x) $ satisfies
\begin{itemize}
\item[i)] $p(ts,x) =t^{-\frac{d}{\alpha }}p( s,t^{-\frac{1}{\alpha }}x)$,

\item[ii)] $p(t,x) \leq p( t,y) $  when $\vert x\vert \geq
\vert y\vert $,

\item[iii)] $p( t,x) \geq (\frac{s}{t}) ^{\frac{d}{\alpha }}p( s,x) $ 
 for $t\geq s$,
 
\item[iv)] $p(t,\frac{1}{\tau }( x-y)) \geq p( t,x)p( t,y) $ 
if $p( t,0) \leq 1$ and $\tau \geq 2$. 
\end{itemize}
\end{lemma}

\begin{proof}
See Guedda and Kirane \cite{G-K} or Sugitani \cite{Su}.
\end{proof}

\begin{lemma}\label{Lemma2}
If there exist $T_{0}>0$ such that
$u(t)=\infty$ or $v(t)=\infty$ for $t\geq T_{0}$, then the
nonnegative solution of \eqref{general} blows up in finite time.
\end{lemma}

\begin{proof}
Due to (\ref{condk}) and Lemma \ref{Lemma1} i), we can assume that
\begin{equation*}
p(K(t,0),0)\leq1 \quad \mbox{ for all }t\geq T_{0}.
\end{equation*}
If $T_{0}\leq \varepsilon _{1}^{1/\rho }t$ and $\varepsilon
_{1}^{1/\rho }t\leq r\leq(2\varepsilon _{1})^{1/\rho }t$, we have
from the conditions of $k(t)$,
\begin{align*}
\tau&\equiv \Big[\frac{K((10\varepsilon _{2})^{1/\rho
}t,r)}{K(r,0) }\Big]^{1/\alpha }=\Big[\frac{K((10\varepsilon
_{2})^{1/\rho
}t,0)-K(r,0)}{K(r,0) }\Big]^{1/\alpha }\\
&\geq \Big[\frac{K((10\varepsilon _{2})^{1/\rho
}t,0)}{K((2\varepsilon _{1})^{1/\rho }t),0) }-1\Big]^{1/\alpha
}\geq\Big[\frac{\varepsilon _{1}(10\varepsilon
_{2})t^\rho}{\varepsilon _{2}(2\varepsilon _{1})t^\rho
}-1\Big]^{1/\alpha }\geq2.
\end{align*}
Hence, using Lemma \ref{Lemma1} i), iv) with
$\tau=\big[\frac{K((10\varepsilon _{2})^{1/\rho }t,r)}{K(r,0)
}\big]^{1/\alpha }$,
\begin{align*}
& p(K((10\varepsilon _{2})^{1/\rho }t,r),x-y)\\
&=p(K(r,0)\Big[\frac{K((10\varepsilon _{2})^{1/\rho
}t,r)}{K(r,0) }\Big],x-y)\\
&=\Big[\frac{K(r,0)}{K((10\varepsilon
_{2})^{1/\rho }t,r) }\Big]^{d/\alpha
}p(K(r,0),\Big[\frac{K(r,0)}{K((10\varepsilon _{2})^{1/\rho }t,r)
}\Big]^{1/\alpha
}(x-y))\\
&\geq\Big[\frac{K(r,0)}{K((10\varepsilon _{2})^{1/\rho }t),r)
}\Big]^{d/\alpha }p(K(r,0),x)p(K(r,0),y), \quad x,y\in
\mathbb{R}^{d}.
\end{align*}
Hence, assuming that $u(t)=\infty$ for all $t\geq T_{0}$,
\begin{equation}
\begin{aligned}
& \int_{\mathbb{R}^{d}}p(K((10\varepsilon _{2})^{1/\rho
}t,r),x-y)u(r,y)dy\\ &\geq \Big[\frac{K(r,0)}{K((10\varepsilon
_{2})^{1/\rho }t,r) }\Big]^{d/\alpha
}p(K(r,0),x)u(r)=\infty,\quad x\in \mathbb{R}^{d}.
\end{aligned}\label{estLemma2}
\end{equation}
We know by (\ref{mildsol2}) that
\begin{align*}
v(t,x)&
=\int_{\mathbb{R}^{d}}p(K(t,0),x-y)\varphi_{2}(y)dy\\
&\quad+\int_{0}^{t}\Big(\int_{\mathbb{R}^{d}}p(K(t,r),x-y)
u^{\beta_{2}}(r,y)dy\Big)dr\\
&\geq\int_{0}^{t}\Big(\int_{\mathbb{R}^{d}}p(K(t,r),x-y)
u^{\beta_{2}}(r,y)dy\Big)dr.
\end{align*}%
Thus,
\begin{equation*}
v((10\varepsilon _{2})^{1/\rho
}t,x)\geq\int_{0}^{(10\varepsilon _{2})^{1/\rho
}t}\Big(\int_{\mathbb{R}^{d}}p(K((10\varepsilon
_{2})^{1/\rho }t,r),x-y)u^{\beta_{2}}(r,y)dy\Big)dr
\end{equation*}
and by Jensen's inequality and (\ref{estLemma2}), we get
\begin{equation*}
v((10\varepsilon _{2})^{1/\rho }t,x)\geq\int_{\varepsilon
_{1}^{1/\rho }t}^{(2\varepsilon _{1})^{1/\rho
}t}\Big(\int_{\mathbb{R}^{d}}p(K((10\varepsilon
_{2})^{1/\rho }t,r),x-y)u(r,y)dy\Big)^{\beta_{2}}dr=\infty,
\end{equation*}
so that $v(t,x)=\infty$ for any $t\geq(10\frac{\varepsilon
_{2}}{\varepsilon _{1}})^{1/\rho }T_{0}$ and $x\in\mathbb{R}^{d}$.
Similarly, when $v(t)=\infty$ for all $t\geq T_{0}$, it can be
shown that $u(t,x)=\infty$ for all $t\geq(10\frac{\varepsilon
_{2}}{\varepsilon _{1}})^{1/\rho }T_{0}$ and $x\in\mathbb{R}^{d}$.
\end{proof}

\begin{theorem}
\label{Thm} If $0< d\rho( \beta_{i} -1)/ \alpha <1$,
$i=1$, $2$, then the nonnegative solution
of system \eqref{general} blows up in finite time.
\end{theorem}

\begin{proof}
Let $t_{0}\geq1$ be such that (\ref{condk}) holds for all $t\geq
t_{0}$ and such that $p(K(t_{0},0),0)\leq1$. Using Lemma
\ref{Lemma1} i), iv), we have
\begin{align*}
p(K(t_{0},0),x-y)=&p(K(t_{0},0),\frac{1}{2}(2x-2y))\geq
p(K(t_{0},0),2x)p(K(t_{0},0),2y)\\
=&2^{-d}p(2^{-\alpha}K(t_{0},0),x)p(K(t_{0},0),2y),\quad x,y\in
\mathbb{R}^{d}.
\end{align*}
Therefore (see (\ref{mildsol1}))
\begin{equation}
\begin{aligned}
u(t_{0},x)
&\geq\int_{\mathbb{R}^{d}}p(K(t_{0},0),x-y)\varphi_{1}(y)dy \\
&\geq2^{-d}p(2^{-\alpha}K(t_{0},0),x)\int_{\mathbb{R}^{d}}p(K(t_{0},0),2y)\varphi_{1}(y)dy\\
&=N_{1}p(2^{-\alpha}K(t_{0},0),x),\quad x\in \mathbb{R}^{d},
\end{aligned}\label{estuto}%
\end{equation}
where
$N_{1}=2^{-d}\int_{\mathbb{R}^{d}}p(K(t_{0},0),2y)\varphi_{1}(y)dy$.
Notice that
\begin{align*}
u(t+t_{0},x)&=\int_{\mathbb{R}^{d}}p(K(t+t_{0},0),x-y)\varphi_{1}(y)dy\\
&\quad+\int_{0}^{t+t_{0}}\Big(\int_{\mathbb{R}^{d}}p(K(t+t_{0},r),x-y)v^{\beta_{1}}
(r,y)dy\Big)dr\\
&=\int_{\mathbb{R}^{d}}p(K(t+t_{0},t_{0})+K(t_{0},0),x-y)\varphi_{1}(y)dy\\
&\quad+\int_{0}^{t_{0}}\Big(\int_{\mathbb{R}^{d}}p(K(t+t_{0},t_{0})+K(t_{0},r),x-y)v^{\beta_{1}}
(r,y)dy\Big)dr\\
&\quad+\int_{t_{0}}^{t+t_{0}}\Big(\int_{\mathbb{R}^{d}}p(K(t+t_{0},r),x-y)v^{\beta_{1}}
(r,y)dy\Big)dr\\
&=\int_{\mathbb{R}^{d}}\Big(\int_{\mathbb{R}^{d}}p(K(t+t_{0},t_{0}),x-z)p(K(t_{0},0),z-y)
dz\Big)\varphi_{1}(y)dy\\
&\quad+\int_{0}^{t_{0}}\Big[\int_{\mathbb{R}^{d}}\Big(\int_{\mathbb{R}^{d}}
p(K(t+t_{0},t_{0}),x-z)p(K(t_{0},r),z-y)dz\Big)\\
&\quad \times v^{\beta_{1}}(r,y)dy\Big]dr\\
&\quad+\int_{0}^{t}\Big(\int_{\mathbb{R}^{d}}p(K(t+t_{0},r+t_{0}),x-y)v^{\beta_{1}}
(r+t_{0},y)dy\Big)dr,\\
\end{align*}
$t\geq0$, $x\in\mathbb{R}^{d}$. From here, by Fubini's
theorem and (\ref{mildsol1}) we have
\begin{align*}
u(t+t_{0},x)&=\int_{\mathbb{R}^{d}}p(K(t+t_{0},t_{0}),x-z)\Big(\int_{\mathbb{R}^{d}}
p(K(t_{0},0),z-y)\varphi_{1}(y)dy\Big)dz\\
&\quad+\int_{\mathbb{R}^{d}}p(K(t+t_{0},t_{0}),x-z)\Big[\int_{0}^{t_{0}}
\Big(\int_{\mathbb{R}^{d}}p(K(t_{0},r),z-y)\\
&\quad \times v^{\beta_{1}}(r,y)dy\Big)dr\Big]dz\\
&\quad+\int_{0}^{t}\Big(\int_{\mathbb{R}^{d}}p(K(t+t_{0},r+t_{0}),x-y)v^{\beta_{1}}
(r+t_{0},y)dy\Big)dr\\
&=\int_{\mathbb{R}^{d}}p(K(t+t_{0},t_{0}),x-y)u(t_{0},y)dy\\
&\quad+\int_{0}^{t}\Big(\int_{\mathbb{R}^{d}}p(K(t+t_{0},r+t_{0}),x-y)v^{\beta_{1}}
(r+t_{0},y)dy\Big)dr,
\end{align*}
$t\geq0,\text{ } x\in\mathbb{R}^{d}$. Thus, using (\ref{estuto})
gives
\begin{equation}
\begin{aligned}
u(t+t_{0},x) &\geq
N_{1}\int_{\mathbb{R}^{d}}p(K(t+t_{0},t_{0}),x-y)p(2^{-\alpha}
K(t_{0},0),y)dy\\
&\quad+\int_{0}^{t}\Big(\int_{\mathbb{R}^{d}}p(K(t+t_{0},r+t_{0}),x-y)
v^{\beta_{1}}(r+t_{0},y)dy\Big)dr\\
&=N_{1}p(K(t+t_{0},t_{0})+2^{-\alpha}K(t_{0},0),x)\\
&\quad+\int_{0}^{t}\Big(\int_{\mathbb{R}^{d}}
p(K(t+t_{0},r+t_{0}),x-y)v^{\beta_{1}}(r+t_{0},y)dy\Big)dr,
\end{aligned}\label{estthm}
\end{equation}
$t\geq0$, $x\in\mathbb{R}^{d}$. Multiplying both sides of
(\ref{estthm}) by $p(K(t+t_{0},0),x)$ and integrating, we have
\begin{align*}
u(t+t_{0})&=
N_{1}p(2K(t+t_{0},t_{0})+(2^{-\alpha}+1)K(t_{0},0),0)\\
&\quad+\int_{0}^{t}\Big(\int_{\mathbb{R}^{d}}
p(2K(t+t_{0},0)-K(r+t_{0},0),y)v^{\beta_{1}}(r+t_{0},y)dy\Big)dr,
\end{align*}
$t\geq0$. Applying Lemma \ref{Lemma1} i), iii), we get
\begin{align*}
u(t+t_{0})
&\geq
N_{1}[2K(t+t_{0},t_{0})+(2^{-\alpha}+1)K(t_{0},0)]^{-d/\alpha
}p(1,0)\\
&\quad+\int_{0}^{t}\Big(\frac{K(r+t_{0},0)}{2K(t+t_{0},0)}\Big)^{d/\alpha}
v^{\beta_{1}}(r+t_{0})dr.
\end{align*}
For a suitable choice of $\theta>0$ given below, we define
$f_{1}(t)=K^{d/\alpha}(t+t_{0},0)u(t+t_{0}),\text{
}g_{1}(t)=K^{d/\alpha}(t+t_{0},0)v(t+t_{0}),\text{ }t\geq\theta$.
Then
\begin{equation*}
f_{1}(t)\geq
\overline{N}_{1}+2^{-d/\alpha}\int_{\theta}^{t}K^{-d(\beta_{1}-1)/\alpha}(r+t_{0},0)
g_{1}^{\beta_{1}}(r)dr,\quad t\geq\theta,
\end{equation*}%
where
$\overline{N}_{1}=p(1,0)N_{1}\Big[\frac{K(\theta,0)}{2K(\theta+t_{0},0)
+(2^{-\alpha}+1)K(t_{0},0)}\Big]^{d/\alpha}$. Similarly, it can
be shown that
\begin{equation*}
g_{1}(t)\geq
\overline{N}_{2}+2^{-d/\alpha}\int_{\theta}^{t}K^{-d(\beta_{2}-1)/\alpha}(r+t_{0},0)
f_{1}^{\beta_{2}}(r)dr,\quad t\geq\theta,
\end{equation*}%
where
$\overline{N}_{2}=p(1,0)N_{2}\Big[\frac{K(\theta,0)}{2K(\theta+t_{0},0)
+(2^{-\alpha}+1)K(t_{0},0)}\Big]^{d/\alpha}$ with
$N_{2}=2^{-d}\int_{\mathbb{R}^{d}}p(K(t_{0},0),\\2y)\varphi_{2}(y)dy$.

Letting $N=\min\{\overline{N}_{1},\overline{N}_{2}\}$, we get
\begin{align*}
f_{1}(t)&\geq
N+2^{-d/\alpha}\int_{\theta}^{t}\min_{i\in \left\{
1,2\right\}
}K^{-d(\beta_{i}-1)/\alpha}(r+t_{0},0)g_{1}^{\beta_{1}}(r)dr,\quad
t\geq\theta,\\
g_{1}(t)&\geq
N+2^{-d/\alpha}\int_{\theta}^{t}\min_{i\in \left\{
1,2\right\}
}K^{-d(\beta_{i}-1)/\alpha}(r+t_{0},0)f_{1}^{\beta_{2}}(r)dr,\quad
t\geq\theta.
\end{align*}
Let $(f_{2}(t),g_{2}(t))$ be the solution of the system integral
equations
\begin{align*}
f_{2}(t)&=
N+2^{-d/\alpha}\int_{\theta}^{t}\min_{i\in \left\{
1,2\right\}
}K^{-d(\beta_{i}-1)/\alpha}(r+t_{0},0)g_{2}^{\beta_{1}}(r)dr,\quad
t\geq\theta,\\
g_{2}(t)&=
N+2^{-d/\alpha}\int_{\theta}^{t}\min_{i\in \left\{
1,2\right\}
}K^{-d(\beta_{i}-1)/\alpha}(r+t_{0},0)f_{2}^{\beta_{2}}(r)dr,\quad
t\geq\theta,
\end{align*}
whose differential expression is
\begin{equation} \label{estthm2}
\begin{aligned}
f_{2}'(t)&= 2^{-d/\alpha}\min_{i\in \left\{ 1,2\right\}
}K^{-d(\beta_{i}-1)/\alpha}(t+t_{0},0)g_{2}^{\beta_{1}}(t),\quad
t>\theta,\\
g_{2}'(t)&= 2^{-d/\alpha}\min_{i\in \left\{ 1,2\right\}
}K^{-d(\beta_{i}-1)/\alpha}(t+t_{0},0)f_{2}^{\beta_{2}}(t),\quad
t>\theta,\\
f_{2}(\theta)&= N,\quad g_{2}(\theta)=N.
\end{aligned}
\end{equation}
 From (\ref{estthm2}) it follows that
\begin{equation*}
\int_{\theta}^{t}f_{2}^{\beta_{2}}(r)f_{2}'(r)dr=\int_{\theta}^{t}
g_{2}^{\beta_{1}}(r)g_{2}'(r)dr,
\end{equation*}
that is,
\begin{equation*}
\frac{1}{\beta_{2}+1}[f_{2}^{\beta_{2}+1}(t)-N^{\beta_{2}+1}]
=\frac{1}{\beta_{1}+1}[g_{2}^{\beta_{1}+1}(t)-N^{\beta_{1}+1}].
\end{equation*}%
Fix $\theta>0$ such that $0<N\leq1$. This is possible due to
$\overline{N}_{1},\text{ }\overline{N}_{2}\to 0$ when $\theta\to
0$. We assume without loss of generality that
$\beta_{2}\geq\beta_{1}$. Then
\begin{equation*}
\frac{f_{2}^{\beta_{2}+1}(t)}{\beta_{2}+1}\leq\frac{g_{2}^{\beta_{1}+1}(t)}{\beta_{1}+1}
\end{equation*}%
or, equivalently
\begin{equation*}
g_{2}(t)
\geq\Big(\frac{\beta_{1}+1}{\beta_{2}+1}\Big)^{\frac{1}{\beta_{1}+1}}
f_{2}^\frac{\beta_{2}+1}{\beta_{1}+1}(t), \quad t\geq\theta.
\end{equation*}
Substituting this in the first equation of (\ref{estthm2}), we
have
\begin{equation*}
f_{2}'(t) \geq2^{-d/\alpha}\min_{i\in \left\{ 1,2\right\}
}K^{-\frac{d(\beta_{i}-1)}{\alpha}}(t+t_{0},0)\Big(\frac{\beta_{1}+1}
{\beta_{2}+1}\Big)^{\frac{\beta_{1}}{\beta_{1}+1}}
f_{2}^{\frac{\beta_{1}(\beta_{2}+1)}{\beta_{1}+1}}(t),\quad
t\geq\theta,
\end{equation*}
that is,
\begin{equation*}
f_{2}^\frac{-\beta_{1}(\beta_{2}+1)}{\beta_{1}+1}(t)f_{2}'(t)
\geq2^{-d/\alpha}\Big(\frac{\beta_{1}+1}{\beta_{2}+1}\Big)^{\frac{\beta_{1}}{\beta_{1}+1}}\min_{i\in
\left\{ 1,2\right\}
}K^{-\frac{d(\beta_{i}-1)}{\alpha}}(t+t_{0},0),\quad t\geq\theta.
\end{equation*}
Integrating from $\theta$ to $t$ yields
\begin{align*}
&\frac{\beta_{1}+1}{1-\beta_{1}\beta_{2}}
\Big[f_{2}^{\frac{1-\beta_{1}\beta_{2}}{\beta_{1}+1}}(t)-N^{\frac{1-\beta_{1}\beta_{2}}{\beta_{1}+1}}\Big]\\
&\geq2^{-d/\alpha}\Big(\frac{\beta_{1}+1}{\beta_{2}+1}\Big)^{\frac{\beta_{1}}{\beta_{1}+1}}
\int_{\theta}^{t}\min_{i\in \left\{ 1,2\right\}
}K^{-\frac{d(\beta_{i}-1)}{\alpha}}(r+t_{0},0)dr.
\end{align*}
Thus (remember that $\beta_{1},$ $\beta_{2}>1$)
\begin{equation*}
f_{2}(t)\geq\Big[N^{\frac{1-\beta_{1}\beta_{2}}{\beta_{1}+1}}-2^{-d/\alpha}\Big(\frac{1-\beta_{1}
\beta_{2}}{\beta_{1}+1}\Big)\Big(\frac{\beta_{1}+1}{\beta_{2}+1}\Big)^{\frac{\beta_{1}}{\beta_{1}+1}}H(t)\Big]^
{\frac{\beta_{1}+1}{1-\beta_{1}\beta_{2}}},
\end{equation*}
where
\begin{equation*}
H(t)\equiv\int_{\theta}^{t}\min_{i\in \left\{
1,2\right\} }K^{-\frac{d(\beta_{i}-1)}{\alpha}}(r+t_{0},0)dr,\quad
t\geq\theta.
\end{equation*}
 From (\ref{condk}) we have
\begin{equation*}
H(t)\geq\int_{\theta}^{t}\min_{i\in \left\{
1,2\right\}
}(\varepsilon_{2}(r+t_{0})^{\rho})^{-\frac{d(\beta_{i}-1)}{\alpha}}dr.
\end{equation*}
Using the fact that $0< d\rho\left( \beta_{2} -1\right)/ \alpha
<1$ we get
\begin{align*}
H(t)&\geq\min_{i\in \left\{ 1,2\right\}
}\varepsilon_{2}^{-\frac{d(\beta_{i}-1)}{\alpha}}\int_{\theta}^{t}
(r+t_{0})^{-\frac{d\rho(\beta_{2}-1)}{\alpha}}dr\\
&=\frac{\alpha}{\alpha-d\rho(\beta_{2}-1)}\min_{i\in
\left\{ 1,2\right\}
}\varepsilon_{2}^{-\frac{d(\beta_{i}-1)}{\alpha}}
\Big[(t+t_{0})^{\frac{\alpha-d\rho(\beta_{2}-1)}{\alpha}}
-(\theta+t_{0})^{\frac{\alpha-d\rho(\beta_{2}-1)}{\alpha}}\Big].
\end{align*}
Thus $H(t)\to\infty$ when $t\to\infty$. So, we have that there
exists $T_{0}\geq\theta$ such that $f_{2}(t)=\infty$ for
$t=T_{0}$. By comparison we have
\begin{equation*}
K^{d/\alpha}(t+t_{0},0)u(t+t_{0})=f_{1}(t)\geq
f_{2}(t)=\infty\quad \mbox{for } t=T_{0},
\end{equation*}
which implies by Lemma \ref{Lemma2} that $v(t,x)=\infty$ for all
$t\geq(10\frac{\varepsilon_{2}}{\varepsilon_{1}})^{1/\rho}(T_{0}
+t_{0})$ and $x\in\mathbb{R}^{d}$. 
\end{proof}


\begin{thebibliography}{99}

\bibitem{A-H-V} D. Andreucci, M. A. Herrero, and J. J. L.
Vel\'{a}zquez. Liouville theorems and blow up behaviour in
semilinear reaction diffusion systems. \emph{Ann. Inst. Henri
Poincar\'{e}, Anal. Non Lin\'{e}aire\/} \textbf{14} (1997), No. 1,
1-53.

\bibitem{B-L-W} M. Birkner, J. A. L\'{o}pez-Mimbela, and A.
Wakolbinger. Blow-up of semilinear PDE's at the critical
demension. A probabilistic approach. \emph{Proc. Amer. Math. Soc.}
\textbf{130} (2002), 2431-2442.

\bibitem{De} W. Deng. Global existence and finite time blow up
for a degenerate reaction-diffusion system. \emph{Nonlinear Anal.
Theory Methods Appl.} \textbf{60} (2005), No. 5(A), 977-991.

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

\bibitem{E-L} M. Escobedo and H. A. Levine.
Critical blow up and global existence numbers for a weakly coupled
system of reaction-diffusion equations. \emph{Arch. Ration. Mech.
Anal.} \textbf{129} (1995), No. 1, 47-100.

\bibitem{F-L-U} M. Fila, H. A. Levine, and Y. Uda.
A Fujita-type global existence-global non-existence theorem for a
system of reaction diffusion equations with differing
diffusivities.  \emph{Math. Methods Appl. Sci.} \textbf{17}
(1994), No. 10, 807-835.

\bibitem{G-K} M. Guedda and M. Kirane.
A note on nonexistence of global solutions to a nonlinear integral
equation.  \emph{Bull. Belg. Math. Soc.} \textbf{6} (1999),
491-497.

\bibitem{Gu-Ki} M. Guedda and M. Kirane. Critically for some
evolution equations. \emph{Differ. Equ.} \textbf{37} (2001), No.
4, 540-550.

\bibitem{K-Q} M. Kirane and M. Qafsaoui. Global nonexistence for
the Cauchy problem of some nonlinear reaction-diffusion systems.
\emph{J. Math. Anal. Appl.} \textbf{268} (2002), 217-243.

\bibitem{Ko} Y. Kobayashi. The life span of blow-up solutions
for a weakly coupled system of reaction-diffusion equations.
\emph{Tokyo J. Math.} \textbf{24} (2001), No. 2, 487-498.

\bibitem{L-W} J. A. L\'{o}pez-Mimbela and A. Wakolbinger.
Lenght of Galton-Watson trees and blow-up of semilinear systems.
\emph{J. Appl. Prob.} \textbf{35} (1998), 802-811.

\bibitem{M-H} K. Mochizuki and Q. Huang. Existence and behaviour
of solutions for a weakly coupled system of reaction-diffusion
equations. \emph{Methods Appl. Anal.} \textbf{5} (1998), No. 2,
109-124.

\bibitem{S-T} P. Souplet and S. Tayachi. Optimal condition for
non-simultaneous blow-up in a reaction-diffusion system, \emph{J.
Math. Soc. Japan} \textbf{56}, (2004), No. 2, 571-584.

\bibitem{Su} S. Sugitani. On nonexistence of global solutions
for some nonlinear integral equations. \emph{Osaka J. Math.}
\textbf{12} (1975), 45-51.

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

\bibitem{Zh} S. Zheng. Nonexistence of positive solutions to
a semilinear elliptic system and blow-up estimates for a
reaction-diffusion system. \emph{J. Math. Anal. Appl.}
\textbf{232} (1999), 293-311.

\end{thebibliography}

\end{document}