Abstract:

This article consider, for the following
heat equation
$$\left\{
\begin{array}{ll}
\displaystyle\frac{u_t}{|x|^s}-\Delta_p u=u^q ,\ \ &(x,t) \in \Omega
\times (0,T),\\
u(x,t)= 0 ,&(x,t) \in \partial \Omega\times(0,T) ,\\
u(x,0)=u_0(x) ,& u_0(x)\ge 0,\quad u_0(x)\not\equiv0
\end{array}
\right. $$
the existence of global solution under some conditions and give two sufficient conditions
for the blow up of local solution in finite time, where $\Omega$ is a smooth bounded domain
in $R^N(N >p)$, $0\in \Omega ,\Delta_p u={\rm div}(|\nabla u|^{p-2}\nabla u),0\le s\le 2,
p\ge 2, p-1