The Shigesada-Kawasaki-Teramoto cross-diffusion system beyond detailed balance

TL;DR

This paper proves the existence of global weak solutions for a complex cross-diffusion system modeling multiple species, using a novel entropy approach that relaxes symmetry conditions and demonstrates convergence to equilibrium.

Contribution

It introduces a new logarithmic entropy method that handles heavily nonsymmetric diffusion matrices without the detailed-balance assumption.

Findings

Existence of global weak solutions for arbitrary species number

Improved conditions on diffusion matrix coefficients

Solutions converge to steady state over time

Abstract

The existence of global weak solutions to the cross-diffusion model of Shigesada, Kawasaki, and Teramoto for an arbitrary number of species is proved. The model consists of strongly coupled parabolic equations for the population densities in a bounded domain with no-flux boundary conditions, and it describes the dynamics of the segregation of the population species. The diffusion matrix is neither symmetric nor positive semidefinite. A new logarithmic entropy allows for an improved condition on the coefficients of heavily nonsymmetric diffusion matrices, without imposing the detailed-balance condition that is often assumed in the literature. Furthermore, the large-time convergence of the solutions to the constant steady state is proved by using the relative entropy associated to the logarithmic entropy.