  We study a system of two coupled nonlinear parabolic equations. It
constitutes a variant of the Keller-Segel model for chemotaxis, i.e. it models
the behaviour of a population of bacteria that interact by means of a
signalling substance. We assume an external confinement for the bacteria and a
nonlinear dependency of the chemotactic drift on the signalling substance
concentration. We perform an analysis of existence and long-time behaviour of
solutions based on the underlying gradient flow structure of the system. The
result is that, for a wide class of initial conditions, weak solutions exist
globally in time and converge exponentially fast to the unique stationary state
under suitable assumptions on the convexity of the confinement and the strength
of the coupling.
