Blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

Abstract

We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1\leq p<m$ and $-2(p-1)/(m-1)<\sigma<\infty$. We prove that there are several types of self-similar solutions with respect to the local behavior near the origin, and their existence depends on the magnitude of $\sigma$. In particular, these solutions have different blow-up sets and rates: some of them have $x=0$ as a blow-up point, some other only blow up at (space) infinity. We thus emphasize on the effect of the weight on the specific form of the blow-up patterns of the equation. The present study generalizes previous works by the authors limited to dimension $N=1$ and $\sigma>0$.