Blow up profiles for a reaction-diffusion equation with critical weighted reaction

We classify the blow up self-similar profiles for the following reaction-diffusion equation with weighted reaction u(t) = (u(m))(xx) + vertical bar x vertical bar(sigma) u(m), posed for (x, t) is an element of R x (0, T), with m > 1 and sigma > 0. In strong contrast with the well-studied equation without the weight (that is sigma = 0), on the one hand we show that for sigma > 0 sufficiently small there exist multiple self-similar profiles with interface at a finite point, more precisely, given any positive integer k, there exists delta(k) > 0 such that for sigma is an element of (0, delta(k)), there are at least k different blow up profiles with compact support and interface at a positive point. On the other hand, we also show that for sigma sufficiently large, the blow up self-similar profiles with interface cease to exist. This unexpected balance between existence of multiple solutions and non-existence of any, when sigma > 0 increases, is due to the effect of the presence of the weight vertical bar x vertical bar(sigma), whose influence is the main goal of our study. We also show that for any sigma > 0, there are no blow up profiles supported in the whole space, that is with u(x, t) > 0 for any x is an element of R and t is an element of (0, T). (C) 2019 Elsevier Ltd. All rights reserved.