Analysis of a Population Model with Strong Cross-Diffusion in an Unbounded Domain

We study a parabolic population model in the full space and prove the global in time existence of a weak solution. This model consists of two strongly coupled diffusion equations describing the population densities of two competing species. The system features intrinsic growth, interand intra-specific competition of the species, as well as diffusion, cross-diffusion and self-diffusion, and drift terms related to varying environment quality. The cross-diffusion terms can be large, making the system non-parabolic for large initial data. The method of our proof is a combination of a time semi-discretization, a special entropy symmetrizing the system, and compactness arguments. 2000 Mathematics Subject Classification: 35K55, 35D05, 92D25