Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator (Q2313925)

In this paper the authors rigorously derive a model for heat diffusion in a composite medium in which the different components are separated by thermally active interfaces. A concentrated problem is derived, leading to a non-standard system of PDEs involving a Laplace-Beltrami operator acting on the interface. The authors then prove the well-posedness of the concentrated problem. The key ingredient in this proof is the contraction mapping theorem, used in conjunction with the theory for abstract parabolic problems. Finally, making use of the Lax-Milgram theorem, the exponential convergence (in time) of the solutions of the system to a steady state is proved.