Global attractor for a nonlocal model for biological aggregation

TL;DR

This paper studies the long-term behavior of solutions to a biological aggregation model with social attraction and dispersal, establishing the existence and structure of global attractors under various conditions.

Contribution

It proves the existence of global attractors for both degenerate and non-degenerate diffusion cases, providing regularity results and structural characterization, and demonstrates exponential attractors for smooth kernels.

Findings

Existence of global attractors in degenerate diffusion case.

Complete structural characterization of the attractor in non-degenerate case.

Convergence of solutions to steady states and existence of exponential attractors.

Abstract

We investigate the long term behavior in terms of global attractors, as time goes to infinity, of solutions to a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. We consider the aggregation equation with both degenerate and non-degenerate diffusion in a bounded domain subject to various boundary conditions. In the degenerate case, we prove the existence of the global attractor and derive some optimal regularity results. Furthermore, in the non-degenerate case we give a complete structural characterization of the global attractor, and also discuss the convergence of any bounded solutions to steady states. Finally, the existence of an exponential attractor is also demonstrated for sufficiently smooth kernels in the case of non-degenerate diffusion.