Parabolic PDEs on a fixed domain with evolving subdomains: function spaces and well-posedness

This paper develops the necessary ingredients for the variational approach of initial boundary-value problems of parabolic partial differential equations on a fixed spatial domain containing evolving subdomains. In particular, we introduce function spaces for the variational solution that extend standard Sobolev-Bochner spaces to account for a coefficient associated with the time derivative that may be discontinuous across the evolving interface. We further show the density of smooth functions in these spaces by extending the mollification technique and the Reynolds transport theorem, and establish the corresponding "embedding" theory and an integration by parts formula. Finally, we prove the well-posedness of the space-time variational formulation in the natural setting using the Banach-Necas-Babuska theorem.