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The existence of global weak solutions to the cross-diffusion model ofShigesada, Kawasaki, and Teramoto for an arbitrary number of species is proved.The model consists of strongly coupled parabolic equations for the populationdensities in a bounded domain with no-flux boundary conditions, and itdescribes the dynamics of the segregation of the population species. Thediffusion matrix is neither symmetric nor positive semidefinite. A newlogarithmic entropy allows for an improved condition on the coefficients ofheavily nonsymmetric diffusion matrices, without imposing the detailed-balancecondition that is often assumed in the literature. Furthermore, the large-timeconvergence of the solutions to the constant steady state is proved by usingthe relative entropy associated to the logarithmic entropy.

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