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

TL;DR

This paper investigates a complex chemotaxis model with nonlinear diffusion and overcrowding prevention, proving existence and regularity of solutions, and comparing it to classical models with numerical illustrations.

Contribution

It establishes existence and regularity results for a doubly nonlinear chemotaxis model with overcrowding prevention, extending classical Keller-Segel models.

Findings

Existence of weak solutions proven using Schauder fixed-point and compactness methods.

Weak solutions exhibit local Hölder regularity via intrinsic scaling.

Numerical examples demonstrate the model's behavior and properties.

Abstract

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 H\"older 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.