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_t u=\Delta u^m+|x|^{sigma}u^p, with m>1, 1<= 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.