The Shigesada-Kawasaki-Teramoto cross-diffusion system beyond detailed balance
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.
Keywords
Cite
@article{arxiv.2207.09876,
title = {The Shigesada-Kawasaki-Teramoto cross-diffusion system beyond detailed balance},
author = {Xiuqing Chen and Ansgar Jüngel and Lei Wang},
journal= {arXiv preprint arXiv:2207.09876},
year = {2022}
}