Analysis of a multidimensional parabolic population model with strong cross-diffusion

The global existence of a nonnegative weak solution to a multidimensional parabolic strongly coupled model for two competing species is proved. The main feature of the model is that the diffusion matrix is nonsymmetric and generally not positive definite and that the nondiagonal matrix elements (the cross-diffusion terms) are allowed to be "large." The ideas of the existence proof are a careful approximation of the cross-diffusion terms using finite differences and the use of an entropy inequality yielding a priori estimates.