Classification of blow-up with nonlinear diffusion and localized reaction

We study the behaviour of nonnegative solutions of the reaction–diffusion equation{ut=(um)xx+a(x)upin R×(0,T),u(x,0)=u0(x)in R. The model contains a porous medium diffusion term with exponent m>1m>1, and a localized reaction a(x)upa(x)up where p>0p>0 and a(x)⩾0a(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)/2p0=(m+1)/2, while the Fujita exponent is pc=m+1pc=m+1: if 0pcp>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>mp>m, while it concentrates to a point in the form of a nonlinear flux if p