Abstract:
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 R and on a bounded interval with no-flux boundary
conditions. In the case of the whole line 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.