Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source

In this paper, we introduce a new method for investigating the rate and pro file of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solution s have global blowup and that the rate of blow-up is uniform in all compact subsets of the domain. This results in a fiat blow-up profile, except for a boundary layer, whose thickness vanishes as t approaches the blow-up time T*. In each case, the blow-up rate of \u(t)\(infinity) is precisely determ ined. Furthermore, in many cases, we derive sharp estimates on the size of the boundary layer and on the asymptotic behavior of the solution in the bo undary layer. The size of the boundary layer then decays like root T* - t, and the solution u(t, x) behaves like \u(t)\(infinity) d(x)/root T* - t in the boundary layer, where d is the distance to the boundary. Some Fujita-ty pe critical exponents results are also given for the Cauchy problem. (C) 19 99 Academic Press.