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Classification of blow-up with nonlinear diffusion and localized reaction

Ferreira de Pablo, Raúl and Pablo, Arturo de and Vázquez, Juan Luis (2006) Classification of blow-up with nonlinear diffusion and localized reaction. Journal of Differential Equations, 231 (1). pp. 195-211. ISSN 0022-0396

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Abstract

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.

Item Type:Article
Uncontrolled Keywords:Blow-up; Porous medium equation; Asymptotic behaviour; Localized reaction; Nonlinear boundary conditions
Subjects:Sciences > Mathematics > Differential equations
ID Code:12493
Deposited On:30 Mar 2011 11:27
Last Modified:06 Feb 2014 09:25

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