Self-Diffusion and Cross-Diffusion Equations: $W^{1,p}$-Estimates and Global Existence of Smooth Solutions

Series:
PDE Seminar
Tuesday, December 3, 2013 - 15:00
1 hour (actually 50 minutes)
Location:
Skiles 006
,
University of Tennessee, Knoxville
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 techniquetogether with a new two-parameter scaling argument.The talk is based on my joint work with Luan Hoang (Texas Tech University) and Truyen Nguyen (University of Akron)