| Abstract: |
| In this talk, the existence of time-global bounds of the Sobolev norm of time global solutions for the
following semilinear parabolic equation involving critical Sobolev exponent will be discussed:
$$
\mbox{(P)}\quad
\left\{
\begin{array}{rclll}
\partial_tu&=&\Delta u+u|u|^{p-2}&\mbox{in}&{\Bbb R}^N\times (0,T_m),\
u|_{t=0}&=&u_0&\mbox{in}&{\Bbb R}^N,
\end{array}
\right.
$$
where $N\geq 3$, $u_0\in H^1({\Bbb R}^N)\cap L^\infty({\Bbb R}^N)$ (for the simplicity) and
$T_m$ denotes the maximal existence time of classical solution of (P) and we only consider the case $T_m=\infty$. It is well-known that
in the subcritical case (the case $p0}\|\nabla u(t)\|_2 |
|