Universal bound for global positive solutions of a superlinear parabolic problem

Abstract.

We prove universal a priori estimates of global positive solutions of the parabolic problem $u_t=\Delta u+u^p$ in $\Omega\times(0,\infty)$, $u=0$ on $\partial\Omega\times(0,\infty)$. Here $\Omega$ is a bounded domain in ${\mathbb R}^n$, $n\leq3$, $p>1$ and p < 5 if n=3.