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 spacial 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 blowup 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.