Existence of type II blowup solutions for a semilinear heat equation with critical nonlinearity

摘要

We consider the blowup rate of solutions for a semilinear heat equation u(t) = Delta u + vertical bar u vertical bar(p-1)u, x is an element of Omega subset of R-N, t > 0, with critical power nonlinearity p = (N + 2)/(N - 2) and N >= 3. First we investigate the profiles of backward self-similar solutions by making use of the variational method, and then, by employing the intersection comparison argument with a particular self-similar solution, we derive the criteria of the blowup rate of solutions, assuming the positivity of solutions in backward space-time parabola. In particular, we show the existence of the so-called type II blowup solutions for the Cauchy-Dirichlet problems on suitable shrinking domains in the case N = 3. (c) 2006 Elsevier Inc. All rights reserved.