Some Free Boundary Problems involving Nonlocal Diffusion and Aggregation

TL;DR

This paper reviews recent advances in free boundary problems involving nonlocal diffusion and aggregation, focusing on anomalous diffusion with long-range effects and complex equilibrium behaviors.

Contribution

It highlights new developments in understanding free boundary problems with nonlocal operators and combined diffusion-aggregation dynamics.

Findings

Progress in modeling anomalous diffusion with fractional operators

Insights into long-term equilibria in diffusion-aggregation systems

Identification of challenges in describing complex free boundary behaviors

Abstract

We report on recent progress in the study of evolution processes involving degenerate parabolic equations what may exhibit free boundaries. The equations we have selected follow to recent trends in diffusion theory: considering anomalous diffusion with long-range effects, which leads to fractional operators or other operators involving kernels with large tails; and the combination of diffusion and aggregation effects, leading to delicate long-term equilibria whose description is still incipient.