Long-time behaviour of solutions to a chemotaxis model with volume-filling effect

The Lyapunov functional is constructed for a quasilinear parabolic system which models chemotaxis and takes into account a volume-filling effect. For some typical case it is proved that the omega-limit set of any trajectory consists of regular stationary solutions. Some lower and tipper bounds oil the stationary solutions are found. For a given range of parameters there are stationary solutions which are inhomogeneous in space.