Universal lower bound of the convergence rate of solutions for a semi-linear heat equation with a critical exponent

Abstract

We study the behavior of solutions of the Cauchy problem for a semi-linear heat equation with critical non-linearity in the sense of Joseph and Lundgren. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give a universal lower bound of the convergence rate of solutions for a class of initial data. This rate contains a logarithmic term, which is not contained in the super critical non-linearity case. Proofs are given by a comparison method based on matched asymptotic expansion.