The Keller–Segel Model for Chemotaxis with Prevention of Overcrowding: Linear vs. Nonlinear Diffusion

Abstract

The aim of this paper is to discuss the effects of linear and nonlinear diffusion in the large time asymptotic behavior of the Keller–Segel model of chemotaxis with volume filling effect. In the linear diffusion case we provide several sufficient conditions for the diffusion part to dominate and yield decay to zero solutions. We also provide an explicit decay rate towards self–similarity. Moreover, we prove that no stationary solutions with positive mass exist. In the nonlinear diffusion case we prove that the asymptotic behavior is fully determined by whether the diffusivity constant in the model is larger or smaller than the threshold value $\varepsilon =1$. Below this value we have existence of nondecaying solutions and their convergence (along subsequences) to stationary solutions. For $\varepsilon >1$ all compactly supported solutions are proved to decay asymptotically to zero, unlike in the classical models with linear diffusion, where the asymptotic behavior depends on the initial mass.