Uniform Blow-Up Profiles and Boundary Behavior for Diffusion Equations with Nonlocal Nonlinear Source

Abstract

In this paper, we introduce a new method for investigating the rate and profile of blow-up of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solutions have global blow-up and that the rate of blow-up is uniform in all compact subsets of the domain. This results in a flat blow-up profile, except for a boundary layer, whose thickness vanishes astapproaches the blow-up timeT*. In each case, the blow-up rate of |u(t)|∞is precisely determined. 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 boundary layer. The size of the boundary layer then decays likeT*−t, and the solutionu(t,x) behaves like |u(t)|∞d(x)/T*−tin the boundary layer, wheredis the distance to the boundary. Some Fujita-type critical exponents results are also given for the Cauchy problem.