Cross diffusion and nonlinear diffusion preventing blow up in the Keller-Segel model

TL;DR

This paper proves global existence and stability of solutions in a modified Keller-Segel model with nonlinear and cross-diffusion terms, preventing blow-up and ensuring boundedness in multiple dimensions.

Contribution

It introduces a new entropy functional for the Keller-Segel model with nonlinear and cross-diffusion, establishing global solutions and stability results.

Findings

Global existence of weak solutions proven for small cross-diffusion coefficients

L^ bounds established for solutions in the parabolic-elliptic case

Numerical experiments confirm theoretical stability and boundedness

Abstract

A parabolic-parabolic (Patlak-) Keller-Segel model in up to three space dimensions with nonlinear cell diffusion and an additional nonlinear cross-diffusion term 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 concentration. For arbitrarily small cross-diffusion coefficients and for suitable exponents of the nonlinear diffusion terms, the global-in-time existence of weak solutions is proved, thus preventing finite-time blow up of the cell density. The global existence result also holds for linear and fast diffusion of the cell density in a certain parameter range in three dimensions. Furthermore, we show $L^\infty$ bounds for the solutions to the parabolic-elliptic system. Sufficient conditions leading to the asymptotic stability of the constant steady state are given for a particular choice of the nonlinear diffusion exponents. Numerical experiments in two and three space dimensions illustrate the theoretical results.