Asymptotic behaviour of solutions of a quasilinear parabolic equation with Robin boundary condition

Abstract

In this paper we study solutions of the quasi-linear parabolic equations $\partial u/\partial t -{\Delta} _p u = a(x) |u|^{q-1}u$ in $(0,T) \times {\Omega} $ with Robin boundary condition ${\partial} u /{\partial} \nu|\nabla u|^{p-2} = b(x) |u|^{r-1}u$ in $(0,T) \times {\partial} {\Omega}$ where $\Omega$ is a regular bounded domain in ${\mathbb R}^N$, $N \geq 3$, $q>1$, $r>1$ and $p \geq 2$. Some sufficient conditions on $a$ and $b$ are obtained for those solutions to be bounded or blowing up at a finite time. Next we give the asymptotic behavior of the solution in special cases.