On a doubly nonlinear diffusion model of chemotaxis with prevention of overcrowding

• Published in: Mathematical methods in the applied sciences Vol. 32; no. 13; pp. 1704 - 1737
• Main Authors: Bendahmane, Mostafa, Bürger, Raimund, Ruiz-Baier, Ricardo, Urbano, José Miguel
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 15.09.2009; Wiley
• Subjects: chemotaxis; degenerate PDE; doubly nonlinear; Exact sciences and technology; Global analysis, analysis on manifolds; intrinsic scaling; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; parabolic p-Laplacian; Partial differential equations; reaction-diffusion equations; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Existence of solution; Numerical linear algebra; parabolic p-Laplacian; Mathematical method; Reaction diffusion equation; degenerate PDE; Partial differential equation; doubly nonlinear; P laplacian; Numerical analysis; Fix point; Weak solution; Applied mathematics; Non linear model; Solution regularity; intrinsic scaling; Condition number; Nonlinearity; reaction-diffusion equations; chemotaxis
• ISSN: 0170-4214; 1099-1476; 1099-1476
• DOI: 10.1002/mma.1107

This paper addresses the existence and regularity of weak solutions for a fully parabolic model of chemotaxis, with prevention of overcrowding, that degenerates in a two‐sided fashion, including an extra nonlinearity represented by a p‐Laplacian diffusion term. To prove the existence of weak solutions, a Schauder fixed‐point argument is applied to a regularized problem and the compactness method is used to pass to the limit. The local Hölder regularity of weak solutions is established using the method of intrinsic scaling. The results are a contribution to showing, qualitatively, to what extent the properties of the classical Keller–Segel chemotaxis models are preserved in a more general setting. Some numerical examples illustrate the model. Copyright © 2008 John Wiley & Sons, Ltd.