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 W1,p-estimates of Calderón-Zygmund type for a class of nonlinear reaction-diffusion equations with self-diffusion. These estimates are achieved by employing the Caffarelli-Peral perturbation technique together with a new two-parameter scaling argument.