Classification of blow-up with nonlinear diffusion and localized reaction

We study the behaviour of nonnegative solutions of the reaction–diffusion equation 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 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 p0=(m+1)/2, while the Fujita exponent is pc=m+1: if 0