Classification of type I and type II behaviors for a supercritical nonlinear heat equation

We study blow-up of radially symmetric solutions of the nonlinear heat equation ut = u + |u|p−1u either on RN or on a finite ball under the Dirichlet boundary conditions.We assumep >pS := N+2 N−2 and that the initial data is bounded, possibly sign-changing. Our first goal is to establish various characterizations of type I and type II blow-ups. Among many other things we show that the following conditions are equivalent: (a) the blow-up is of type II; (b) the rescaled solution w(y, s) converges to either ϕ ∗ (y) or −ϕ ∗ (y) as s→∞, where ϕ ∗ denotes the singular stationary solution; (c) u(x, T )/ϕ ∗ (x) tends to ±1 as x→0, where T is the blow-up time. Our second goal is to study continuation beyond blow-up. Among other things we show that if a blow-up is of type I and incomplete, then its limit L1 continuation becomes smooth immediately after blow-up, and that type I blow-up implies “type I regularization,” that is, (t −T )1/(p−1) u(·, t) L ∞ is bounded as t T . We also give various criteria for complete and incomplete blow-ups. © 2008 Elsevier Inc. All rights reserved.