Blowup behavior of solutions for a semilinear heat equation with supercritical nonlinearity

This paper is concerned with blowup phenomena of solutions for the Cauchy and the Cauchy-Dirichlet problem of u(1)= Deltau+u(p) with p in the supercritical range in the sense of Sobolev's embedding. We first show that if p> 1 + 7/(N - 11) and Ngreater than or equal to12, then there are no radially symmetric bounded positive solutions of Deltaw- y/2 delw - 1/p-1 w+w(p) = 0 in R-N which intersect the radially symmetric singular solution at least twice. Using the above result, the existence of a blowup solution of type II for the Cauchy-Dirichlet problem for (P) in a ball is proved, where a solution u is said to exhibit the type II blowup at t = T if lim sup(tNE arrowT) (T- t)(1/(p-1)) \u(t)\(infinity) = infinity. (C) 2004 Elsevier Inc. All rights reserved.