I will present joint work with C. Bandle, M. del Mar Gonzalez and M. Fontelos on different asymptotic questions for a nonlocal diffusion problem determined by a smooth kernel of compact support.

The main questions are if and how does the geometry of the manifold influence different kinds of asymptotic problems such as the infinitesimal limit as the support of the kernel contracts to 0 or the large time asymptotic behavior of the solution for a fixed kernel.