  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.
