Abstract. The aim of this paper is to discuss the effects of linear and nonlinear diffusion in the Keller-Segel model of chemotaxis with volume filling effect. In both cases we first cover the global existence and uniqueness theory of solutions of the Cauchy problem on R d . Then, we address the large time asymptotic behavior. In the linear diffusion case we provide several sufficient conditions such that the diffusion part dominates and yields decay to zero of 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 behaviour is fully determined by the diffusivity constant in the model being larger or smaller than the threshold value ε = 1. Below this value we have existence of non-decaying solutions and their convergence (in terms of subsequences) to stationary solutions. For ε > 1 all compactly supported solutions are proven to decay asymptotically to zero, unlike in the classical models with linear diffusion, where the asymptotic behaviour depends on the initial mass.
Burger et al. REF prove the global existence and uniqueness of the Cauchy problem in R N for linear and nonlinear diffusion with prevention of overcrowding.
The Keller-Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion
SIAM Journal on Mathematical Analysis
Mathematics Computer Science