Large-time asymptotics for degenerate cross-diffusion population models with volume filling

TL;DR

This paper investigates the long-term behavior of solutions to complex degenerate cross-diffusion models describing multiple interacting populations, introducing novel entropy methods to handle degeneracies and different diffusivities.

Contribution

It extends previous analyses by allowing for varying diffusivities and degenerate nonlinearities, employing convex Sobolev inequalities with modified entropy densities.

Findings

Established large-time asymptotics for the models

Developed new entropy-based techniques for degenerate systems

Handled multiple species with different diffusion properties

Abstract

The large-time asymptotics of the solutions to a class of degenerate parabolic cross-diffusion systems is analyzed. The equations model the interaction of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. Compared to previous works, we allow for different diffusivities and degenerate nonlinearities. The proof is based on the relative entropy method, but in contrast to usual arguments, the relative entropy and entropy production are not directly related by a logarithmic Sobolev inequality. The key idea is to apply convex Sobolev inequalities to modified entropy densities including "iterated degenerate" functions.