Analysis of a parabolic cross-diffusion population model without self-diffusion

Abstract

The global existence of non-negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown. The cross-diffusion terms are allowed to be arbitrarily large, whereas the self-diffusion terms are assumed to disappear. The last assumption complicates the analysis since these terms usually provide H 1 estimates of the solutions. The existence proof is based on a positivity-preserving backward Euler–Galerkin approximation, discrete entropy estimates, and L 1 weak compactness arguments. Furthermore, employing the entropy–entropy production method, we show for special stationary solutions that the transient solution converges exponentially fast to its steady state. As a by-product, we prove that only constant steady states exist if the inter-specific competition parameters disappear no matter how strong the cross-diffusion constants are.