Gradient Estimates and Global Existence of Smooth Solutions to a Cross-Diffusion System

We investigate the global time existence of smooth solutions for the Shigesada-Kawasaki-Teramoto system of cross-diffusion equations of two competing species in population dynamics. If there are self-diffusion in one species and no cross-diffusion in the other, we show that the system has a unique smooth solution for all time in bounded domains of any dimension. We obtain this result by deriving global $W^{1,p}$-estimates of Calder\'{o}n-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument. Lausanne, le 2 juin 2014 BD/HMN/MM _______ Les séminaires qui ont lieu à la Section de Mathématiques sont annoncés sur Internet