Asymptotic behaviour of the heat equation in an exterior domain with general boundary conditions II. The case of bounded and of $L^{p}$ data

TL;DR

This paper investigates the long-term behavior of heat equation solutions in exterior domains with various boundary conditions, focusing on bounded and $L^p$ initial data, revealing how solutions approximate whole-space solutions or decay to zero.

Contribution

It extends previous work by analyzing the asymptotic behavior for bounded and $L^p$ initial data, including correction terms and decay rates, under different boundary conditions.

Findings

Solutions with bounded data approach whole-space solutions after correction.

For $L^p$ data, solutions decay to zero with potentially slow convergence.

Complex behaviors emerge depending on initial data and boundary conditions.

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 ones. In this second part of our work, we consider the case of bounded initial data and prove that, after some correction term, the solutions become close to the solutions in the whole space and show how complex behaviours appear. We also analyse the case of initial data in $L^p$ with $1<p<\infty$ where all solutions essentially decay to $0$ and the convergence rate could be arbitrarily slow.