%0 Journal Article
%A Ferreira de Pablo, Raúl
%A Pablo, Arturo de
%A Vázquez, Juan Luis
%T Classification of blow-up with nonlinear diffusion and localized reaction
%D 2006
%@ 0022-0396
%U https://hdl.handle.net/20.500.14352/49642
%X We study the behaviour of nonnegative solutions of the reaction-diffusion equation _ ut = (um)xx + a(x)up in R × (0, T), u(x, 0) = u0(x) in R.The model contains a porous medium diffusion term with exponent m > 1, and a localized reaction a(x)up where p > 0 and a(x) ≥ 0 is a compactly supported 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 p0 = (m + 1)/2, while the Fujita exponent is pc = m + 1: if 0 < p ≤ p0 every solution is global in time, if p0 < p ≤ pc all solutions blow up and if p > pc 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.
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