Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

Abstract

We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: ∂tu = ∆u^m + |x|^σu^p, posed in any space dimension x ∈ R^N , t ≥ 0 and with exponents m > 1, p ∈ (0, 1) and σ > 2(1−p)/(m−1). We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are compactly supported and might present two different types of interface behavior and three different possible good behaviors near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of σ. This paper generalizes in dimension N > 1 previous results by the authors in dimension N = 1 and also includes some finer classification of the profiles for σ large that is new even in dimension N = 1.