We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. When diffusion is stronger than convec- tion the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a “projection” of the original one onto the subspace. The determination of this projection is one of the main issues. When convection and diffusion are of the same order the limit equation coincides with the original one.

We are able to cover both the cases of slow and fast convection, as long as the mass is preserved.

This is a joint work with Jørgen Endal (Norwegian University of Science and Technology, Trondheim, Norway) and Fernando Quirós (Universidad Autonoma de Madrid, Spain).