Communications in Mathematical Sciences

Volume 14 (2016)

Number 1

Stationary solutions with vacuum for a one-dimensional chemotaxis model with nonlinear pressure

Pages: 147 – 186

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n1.a6

Authors

Florent Berthelin (Laboratoire J. A. Dieudonné, UMR 7351, CNRS-Université Nice-Sophia Antipolis, Parc Valrose, Nice, France; and Project Team COFFEE, INRIA Sophia Antipolis, France)

David Chiron (Laboratoire J. A. Dieudonné, UMR 7351, CNRS-Université Nice-Sophia Antipolis, Parc Valrose, Nice, France; and Project Team COFFEE, INRIA Sophia Antipolis, France)

Magali Ribot (Laboratoire J. A. Dieudonné, UMR 7351, CNRS-Université Nice-Sophia Antipolis, Parc Valrose, Nice, France; and Project Team COFFEE, INRIA Sophia Antipolis, France)

Abstract

In this article, we study a one-dimensional hyperbolic quasilinear model of chemotaxis with a nonlinear pressure and we consider its stationary solutions, in particular with vacuum regions. We study both cases of the system set on the whole line $\mathbb{R}$ and on a bounded interval with no-flux boundary conditions. In the case of the whole line $\mathbb{R}$, we find only one stationary solution, up to a translation, formed by a positive density region (called bump) surrounded by two regions of vacuum. However, in the case of a bounded interval, an infinite of stationary solutions exists, where the number of bumps is limited by the length of the interval. We are able to compare the value of an energy of the system for these stationary solutions. Finally, we study the stability of these stationary solutions through numerical simulations.

Keywords

chemotaxis, stationary solutions, vacuum, quasilinear hyperbolic problem, energy

2010 Mathematics Subject Classification

35B35, 35L60, 35Q92, 92D25

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Published 16 September 2015