Type II blow-up mechanisms in a semilinear heat equation with Lepin exponent

Abstract

We are concerned with blow-up mechanisms in a semilinear heat equation:ut=Δu+|u|p−1u,x∈RN,t>0, where p>1 is a constant. In the present article, we prove constructively the existence of type II blow-up solutions for p=1+6/(N−10), so called Lepin exponent. Their blow-up mechanisms are strongly influenced on space dimensions. For lower dimensions the mechanisms are determined by Dirac δ type approximations of nonlinear term and, on the other hand, dominated by classical Taylor approximation for high dimensions. On the threshold dimension, the both approximations are balanced. Consequently, the blow-up exhibits a new blow-up profile. Classification of radial blow-up solutions in terms of blow-up rates is also presented.