  In this article we study a non-local diffusion problem that involves three
different fractional Laplacian operators acting on two domains. Each domain has
an associated operator that governs the diffusion on it, and the third operator
serves as a coupling mechanism between the two of them. The model proposed is
the gradient flow of a non-local energy functional. In the first part of the
article we provide results about existence of solutions and the conservation of
mass. The second part encompasses results about the Lp decay of the solutions.
The third part is devoted to study the asymptotic behavior of the solutions of
the problem when the two domains are a ball and its complementary. Exterior
fractional Sobolev and Nash inequalities of independent interest are also
provided in an appendix.
