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.