BLOW-UP PATTERNS FOR A REACTION-DIFFUSION EQUATION WITH WEIGHTED REACTION IN GENERAL DIMENSION

We classify all the blow-up solutions in self -similar form to the following reaction-diffusion equation partial derivative(t)u= triangle u(m)+|x|(sigma)u(p), posed for (x, t) is an element of R(N )x (0,T), with m >1, 1 <= p < m and -2(p-1)/(m-1) < sigma < infinity. 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.