Classification of blow-up with nonlinear diffusion and localized reaction

We study the behaviour of nonnegative solutions of the reaction-diffusion equation u(t) = (u(m))(xx) + a(x)u(p) in R x (0, T), u (x, 0) = u(0) (x) in R. The model contains a porous medium diffusion term with exponent m > 1, and a localized reaction a(x)u(p) where p > 0 and a(x) >= 0 is a compactly supported symmetric function. We investigate the existence and behaviour of the solutions of this problem in dependence of the exponents m and p. We prove that the critical exponent for global existence is p(0) = (m + 1)/2, while the Fujita exponent is p(c) = m + 1: if 0 < p <= p(0) every solution is global in time, if p(0) < p <= p(c) all solutions blow up and if p > p(c) both global in time solutions and blowing up solutions exist. In the case of blow-up, we find the blow-up rates, the blow-up sets and the blow-up profiles; we also show that reaction happens as in the case of reaction extended to the whole line if p > m, while it concentrates to a point in the form of a nonlinear flux if p < m. If p = m the asymptotic behaviour is given by a self-similar solution of the original problem. (c) 2006 Elsevier Inc. All rights reserved.