On blow-up rate for the Hénon parabolic equation with Sobolev supercritical nonlinearity
We discuss the Hénon parabolic equation $\partial_t u = Δu + |x|^σu^p$ in a finite ball in $\mathbb{R}^N$ under the Dirichlet boundary condition, where $N\ge1$, $p>1$, and $σ>0$. We assume that the exponent $p$ is supercritical in the Sobolev sense. Since the spatial potential term $|x|^σ$ vanishes at the origin, solutions seem less likely to blow up at the origin. We construct a solution that blows up at the origin and also carry out an analysis of blow-up rate of solutions. In particular, if $p$ is less than the Joseph--Lundgren exponent, all blow-ups are shown to be of Type I. The lower bound corresponding to Type I rate is also shown for some particular blow-up solutions. As by products, we present a basic result on classification to threshold solutions for every $p>1+σ/N$.