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

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 smooth compact set in \({\mathbb {R}}^N\) ${\mathbb{R}}^N$ . Different homogeneous boundary conditions are considered, including Dirichlet, Robin, and Neumann conditions. We determine the exact amount of mass lost and identify criteria for complete mass decay, in which the dimension of the space plays a key role. Furthermore, we provide explicit mass decay rates.