Domain perturbation for parabolic equations

Abstract

We study the effect of domain perturbation on the behaviour of parabolic equations. In particular, we show how solutions of parabolic equations behave as a sequence of domains ${\Omega }_n$ in ${ℝ}^N$ converges to an open set $\Omega$ in a certain sense. We are interested in singular domain perturbations so that a change of variables is not possible on these domains. In the first part of this talk, we concentrate on initial-boundary value problems for non-autononous parabolic equations. We prove the convergence of solutions by variational methods using the notion of Mosco convergence. In the second part, we look at domain perturbation for bounded solutions of parabolic equations on the whole real line.