Classification of blow-up with nonlinear diffusion and localized reaction

Abstract

We study the behaviour of nonnegative solutions of the reaction–diffusion equation { u t = ( u m ) x x + a ( x ) u p in R × ( 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.