On the loss of mass for the heat equation in an exterior domain with general boundary conditions

TL;DR

This paper investigates how solutions to the heat equation in exterior domains lose mass over time under various boundary conditions, providing explicit decay rates and criteria for complete mass decay influenced by the spatial dimension.

Contribution

It offers a comprehensive analysis of mass decay in exterior domains with general boundary conditions, including explicit decay rates and criteria for total mass loss, highlighting the role of dimension.

Findings

Explicit mass decay rates derived

Criteria for complete mass decay established

Dimension significantly influences mass loss behavior

Abstract

In this work, we study the decay of mass for solutions to the heat equation in exterior domains, i.e., domains which are the complement of a compact set in $\mathbb{R}^N$. Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann conditions. We determine the exact amount of mass loss and identify criteria for complete mass decay, in which the dimension of the space plays a key role. Furthermore, the paper provides explicit mass decay rates.