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

Florent Berthelin (1, 2) , David Chiron (3) , Magali Ribot (1, 2)

1 LJAD - Laboratoire Jean Alexandre Dieudonné
2 COFFEE - COmplex Flows For Energy and Environment
3 JAD - Laboratoire Jean Alexandre Dieudonné

In this article, we study a one-dimensional hyperbolic quasi-linear model of chemotaxis with a non-linear pressure and we consider its stationary solutions, in particular with vacuum regions. We study both cases of the system set on the whole line $\Er$ and on a bounded interval with no-flux boundary conditions. In the case of the whole line $\Er$, 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.

Domaines: Analyse numérique [math.NA]

Florent Berthelin, David Chiron, Magali Ribot. Stationary solutions with vacuum for a one-dimensional chemotaxis model with non-linear pressure. Communications in Mathematical Sciences, 2016, 14 (1), pp.147-186. ⟨hal-01070625⟩