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.

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We present partial differential equation (PDE) model hierarchies for the chemotactically driven motion of biological cells. Starting from stochastic differential models, we derive a kinetic formulation of cell motion coupled to diffusion equations for the chemoattractants. We also derive a fluid dynamic (macroscopic) Keller-Segel type chemotaxis model 1173 Math. Models Methods Appl. Sci. 2006.16:1173-1197. Downloaded from www.worldscientific.com by UNIVERSITY OF PITTSBURGH on 03/10/15. For personal use only. 1174 F. Chalub et al. by scaling limit procedures. We review rigorous convergence results and discuss finitetime blow-up of Keller-Segel type systems. Finally, recently developed PDE-models for the motion of leukocytes in the presence of multiple chemoattractants and of the slime mold Dictyostelium Discoideum are reviewed.

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