Quenching rate of solutions for a semilinear parabolic equation

Abstract

We study the behavior of solutions of the Cauchy problem for a semilinear parabolic equation with a singular absorption term. We discuss the convergence of solutions to a singular stationary solution from above as time goes to infinity, and show that in a supercritical case a sharp estimate of the quenching rate can be determined explicitly when a specific growth rate of initial data is given. We also obtain a universal lower bound of the quenching rate which implies the optimality of the results. Proofs are given by a comparison method that is based on matched asymptotic expansion. We first determine a quenching rate of solutions by a formal analysis. Based on the formal analysis, we give a rigorous proof by constructing appropriate super- and subsolutions with the desired quenching rate.