The aim of this paper is to discuss the effects of linear and non- linear difusion 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 Rd. Then, we address the large time asymptotic behavior. In the linear diffusion case we provide several su±cient conditions such that the di®usion part dominates and yields decay to zero of solutions. We also provide an explicit decay rate towards self{similarity. More- over, 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 epsilon = 1. Below this value we have existence of non-decaying solutions and their convergence (in terms of subsequences) to stationary solutions. For epsilon > 1 all compactly supported solutions are proven to decay asymptot- ically to zero, unlike in the classical models with linear diffusion, where the asymptotic behaviour depends on the initial mass.

@Article{BDD06,
  author       = {Burger, M. and Di Francesco, M. and Dolak-Struss, Y.},
  title        = {The Keller-Segel model for chemotaxis: linear vs. nonlinear diffusion},
  journal      = {SIAM J. Math. Anal.},
  volume       = {38},
  pages        = {1288-1315.},
  year         = {2006},
  url          = \{/2006/BDD06},
}