On nonexistence of type II blowup for a supercritical nonlinear heat equation

Abstract

In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut = Δu + |u|p-1u either on ℝN or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, P > Ps:= N+2/N-2 We prove that if Ps < P p*, the above range of exponent pis p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > ps and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. We then establish useful estimates for the so-called incomplete blowup, which reveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. © 2004 Wiley Periodicals, Inc.