Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator

In this paper we derive a model for heat diffusion in a composite medium in which the different components are separated by thermally active interfaces. The previous result is obtained via a concentrated capacity procedure and leads to a non-stantard system of PDEs involving a Laplace-Beltrami operator acting on the interface. For such a system well-posedness is proved using contraction mapping and abstract parabolic problems theory. Finally, the exponential convergence (in time) of the solutions of our system to a steady state is proved.