CROSS DIFFUSION PREVENTING BLOW-UP IN THE TWO-DIMENSIONAL KELLER-SEGEL MODEL

A (Patlak-)Keller-Segel model in two space dimensions with an additional cross-diffusion term in the equation for the chemical signal is analyzed. The main feature of this model is that there exists a new entropy functional, yielding gradient estimates for the cell density and chemical substance. This allows one to prove, for arbitrarily small cross diffusion, the global existence of weak solutions to the parabolic-parabolic model as well as the global existence of bounded weak solutions to the parabolic-elliptic model, thus preventing blow-up of the cell density. Furthermore, the long-time decay of the solutions to the parabolic-elliptic model is shown and finite-element simulations are presented illustrating the influence of the regularizing cross-diffusion term.