A non-local coupling model involving three fractional Laplacians

In this paper, 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 paper, 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 Appendix A.