Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection

TL;DR

This paper investigates the long-term behavior of solutions to a nonlocal heat equation with anisotropic diffusion and convection, revealing how the dominant process influences the asymptotic limit and extending results to cases with non-Lipschitz convection.

Contribution

It introduces a comprehensive analysis of large-time limits for anisotropic nonlocal diffusion with convection, including cases with non-Lipschitz nonlinearities, and characterizes the limit equations based on the relative strength of diffusion and convection.

Findings

When diffusion dominates, solutions behave like the purely diffusive equation.

In convection-dominated regimes, the limit equation reflects a projection of the diffusion operator.

Results extend to the isotropic fractional Laplacian case and include non-Lipschitz convection nonlinearities.

Abstract

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 L\'evy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution with this mass of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection 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 of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one. Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally H\"older, has not been considered before in the nonlocal diffusion setting.