Asymptotic behaviour of the heat equation in an exterior domain with general boundary conditions I. The case of integrable data

TL;DR

This paper investigates the long-term behavior of heat equation solutions in exterior domains with various boundary conditions, revealing how mass loss and boundary types influence asymptotic profiles and time decay rates.

Contribution

It provides a comprehensive analysis of the asymptotic behavior of heat solutions with general boundary conditions in exterior domains, including optimal decay rates and mass loss effects.

Findings

Asymptotic profiles depend on boundary conditions and initial data.

Mass loss occurs through the boundary, affecting long-term distribution.

Results include optimal time decay rates for solutions.

Abstract

In this work, we study the asymptotic behaviour of solutions to the heat equation in exterior domains, i.e., domains which are the complement of a smooth compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann conditions for integrable initial data in $L^1(\Omega)$. After taking into account the loss of mass of the solution through the boundary, depending on the boundary conditions, we describe the asymptotic spatial distribution of the remaining mass in terms of the Gaussian and of a suitable asymptotic profile function. We show that our results have optimal time rates.