Stability of the blow-up time and the blow-up set under perturbations

Abstract

In this paper we prove a general result concerning continuity of the blow-up time and the blow-up set for an evolution problem under perturbations. This result is based on some convergence of the perturbations for times smaller than the blow-up time of the unperturbed problem together with uniform bounds on the blow-up rates of the perturbed problems. We also present several examples. Among them we consider changing the special domain in which the heat equation with a power source takes place. We consider rather general perturbations of the domain and show the continuity of the blow- up time. Moreover, we deal with perturbations on the initial condition and on parameters in the equation. Finally, we also present some continuity results for the blow-up set