PARABOLIC AND NAVIER–STOKES EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 81 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2008 CHEMOTAXIS MODELS WITH A THRESHOLD CELL DENSITY DARIUSZ WRZOSEK Institute of Applied Mathematics and Mechanics, Warsaw University Banacha 2, 02-097 Warszawa, Poland E-mail: darekw@mimuw.edu.pl Abstract. We consider a quasilinear parabolic system which has the structure of Patlak-Keller- Segel model of chemotaxis and contains a class of models with degenerate diffusion. A cell pop- ulation is described in terms of volume fraction or density. In the latter case, it is assumed that there is a threshold value which the density of cells cannot exceed. Existence and uniqueness of solutions to the corresponding initial-boundary value problem and existence of space inhomoge- neous stationary solutions are discussed. In the 1D case a classification of stationary solutions for some model example is provided. 1. Introduction. We are dealing with a class of quasilinear systems of parabolic equa- tions which are used to model the chemotactic motion of biological cells. Chemotaxis is understood here as a chemosensitive oriented movement of biological cells which may detect and response to some chemical (chemoattractant) secreted to their environment. It is assumed that the total flux of cells consists of diffusive and chemotactic parts. The classical model describing the aggregation phase of chemosensitive motion of cells was introduced by Patlak [11] and Keller and Segel [6]. Most of works on Patlak/Keller-Segel model were focused on the case when cell diffusion is Brownian and chemotactic sensitiv- ity is a density-independent fixed constant. Recently new models which have nonlineari- ties in both chemotactic and diffusive parts have been introduced by Hillen and Painter [2, 10], and Byrne and Owen [3]. The latter is due to the multiphase modeling and the former takes into account a volume filling effect. For brevity we shall refer to (MP) and (VF) in the sequel. On the one hand, derivation and interpretation of both models are based on different approaches, but on the other one they turn out to share some common 2000 Mathematics Subject Classification : Primary 35K55, 35K65; Secondary 34B15, 34C25. Key words and phrases : chemotaxis equations, degenerate diffusion, quasilinear parabolic equation, compactness method. Supported by Polish KBN grant 2 P03A 03022. The paper is in final form and no version of it will be published elsewhere. [553] c  Instytut Matematyczny PAN, 2008