Large-time estimates for the Dirichlet heat equation in exterior domains

Abstract

We give large-time asymptotic estimates, both in uniform and $L^1$ norms, for solutions of the Dirichlet heat equation in the complement of a bounded open set of $\mathbb{R}^d$ satisfying certain technical assumptions. We always assume that the initial datum has suitable finite moments (often, finite first moment). All estimates include an explicit rate of approach to the asymptotic profiles at the different scales natural to the problem, in analogy with the Gaussian behaviour of the heat equation in the full space. As a consequence we obtain by an approximation procedure the asymptotic profile, with rates, for the Dirichlet heat kernel in these exterior domains. The estimates on the rates are new even when the domain is the complement of the unit ball in $\mathbb{R}^d$, except for previous results by Uchiyama in dimension 2, which we are able to improve in some scales. We obtain that the heat kernel behaves asymptotically as the heat kernel in the full space, with a factor that takes into account the shape of the domain through a harmonic profile, and a second factor which accounts for the loss of mass through the boundary. The main ideas we use come from entropy methods in PDE and probability, whose application seems to be new in the context of diffusion problems in exterior domains.