Exact convergence rate of solutions for a semilinear heat equation with a critical and a supercritical exponent revisited

We study the behavior of solutions of the Cauchy problem for a semilinear heat equation with supercritical and critical nonlinearity 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 the above cases. In this paper, for the supercritical case, we give a lower bound of a convergence rate that leads to the exact convergence rate together with our previous result. Also for the critical case, we give the exact convergence rate of solutions depending on two approaching initial data near spatial infinity again by using a different function than the previous results. For the critical case, this rate contains a logarithmic factor which is not contained in the supercritical nonlinearity case. Proofs are given by a comparison method based on matched asymptotic expansion.