  The global-in-time existence of nonnegative bounded weak solutions to a class
of cross-diffusion systems for two population species is proved. The
diffusivities are assumed to depend linearly on the population densities in
such a way that a certain formal gradient-flow structure holds. The main
feature of these systems is that the diffusion matrix may be neither symmetric
nor positive definite. The key idea of the proof is to employ the
boundedness-by-entropy principle which yields at the same time the existence of
global weak solutions and their boundedness. In particular, the uniform
boundedness of weak solutions to the population model of Shigesada, Kawasaki,
and Teramoto in several space dimensions under certain conditions on the
diffusivities is shown for the first time.
