A nonlocal regularization of a generalized Busenberg-Travis cross-diffusion system

TL;DR

This paper analyzes a nonlocal regularization of a generalized cross-diffusion system modeling segregating populations, establishing existence, uniqueness, decay, and limiting behavior of solutions.

Contribution

It introduces a nonlocal regularization of a Busenberg-Travis model and proves key properties like existence, boundedness, and decay of solutions, including weak-strong uniqueness.

Findings

Existence of global weak solutions

Solutions decay exponentially over time

Weak-strong uniqueness for the limiting system

Abstract

A cross-diffusion system with Lotka-Volterra reaction terms in a bounded domain with no-flux boundary conditions is analyzed. The system is a nonlocal regularization of a generalized Busenberg-Travis model, which describes segregating population species with local averaging. The partial velocities are the solutions of an elliptic regularization of Darcy's law, which can be interpreted as a Brinkman's law. The following results are proved: the existence of global weak solutions; localization limit; boundedness and uniqueness of weak solutions (in one space dimension); exponential decay of the solutions. Moreover, the weak-strong uniqueness property for the limiting system is shown.