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

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.