Convergence rate for a semilinear parabolic equation with a singular nonlinearity

Abstract

We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with a singular power nonlinearity. It is known for a supercritical heat equation that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In fact, given a specific decay rate of initial data, a sharp estimate of the convergence rate can be obtained explicitly. The main purpose of this research is to show a similar result for a semilinear parabolic equation with a singular absorption term. We also obtain a universal lower bound of the convergence rate which implies the optimality of the result. Proofs are given by a comparison method that is based on matched asymptotic expansion.