Separate variable blow-up patterns for a reaction-diffusion equation with critical weighted reaction

We study the separate variable blow-up patterns associated to the following second order reaction-diffusion equation: partial derivative(t)u = Delta u(m) + vertical bar x vertical bar(sigma)u(m), posed for x is an element of R-N, t >= 0, where m > 1, dimension N >= 2 and sigma > 0. A new and explicit critical exponent sigma(c) = 2(m - 1)(N - 1)/3m + 1 is introduced and a classification of the blow-up profiles is given. The most interesting contribution of the paper is showing that existence and behavior of the blow-up patterns is split into different regimes by the critical exponent sigma(c) and also depends strongly on whether the dimension N >= 4 or N is an element of {2, 3}. These results extend previous works of the authors in dimension N = 1. (C) 2021 Elsevier Ltd. All rights reserved.