Ferreira de Pablo, Raúl and Pablo, Arturo de and Vázquez, Juan Luis (2006) Classification of blowup with nonlinear diffusion and localized reaction. Journal of Differential Equations, 231 (1). pp. 195211. ISSN 00220396

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Official URL: http://www.sciencedirect.com/science/journal/00220396
Abstract
We study the behaviour of nonnegative solutions of the reactiondiffusion 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 blowup, we find the blowup rates, the blowup sets and the blowup 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 selfsimilar solution of the original problem.
Item Type:  Article 

Uncontrolled Keywords:  Blowup; 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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