  In this paper, we study some qualitative properties for an evolution problem
that combines local and nonlocal diffusion operators acting in two different
subdomains and, coupled in such a way that, the resulting evolution problem is
the gradient flow of an energy functional. The coupling takes place at the
interface between the regions in which the different diffusions take place. We
prove existence and uniqueness results, as well as, that the model preserves
the total mass of the initial condition. We also study the asymptotic behavior
of the solutions. Finally, we show a suitable way to recover the heat equation
at the whole domain from taking the limit at the nonlocal rescaled kernel.
