  We prove existence, uniqueness and several qualitative properties for
evolution equations that combine local and nonlocal diffusion operators acting
in different subdomains and coupled in such a way that the resulting evolution
equation is the gradient flow of an energy functional. We deal with the Cauchy,
Neumann and Dirichlet problems, in the last two cases with zero boundary data.
For the first two problems we prove that the model preserves the total mass. We
also study the behaviour of the solutions for large times. Finally, we show
that we can recover the usual heat equation (local diffusion) in a limit
procedure when we rescale the nonlocal kernel in a suitable way.

Analysis of PDEs