  This paper presents new analytical results for a class of nonlinear parabolic
systems of partial different equations with small cross-diffusion which
describe the macroscopic dynamics of a variety of large systems of interacting
particles. Under suitable assumptions, we prove existence of classical
solutions and we show exponential convergence in time to the stationary state.
Furthermore, we consider the special case of one mobile and one immobile
species, for which the system reduces to a nonlinear equation of Fokker-Planck
type. In this framework, we improve the convergence result obtained for the
general system and we derive sharper $L^{\infty}$-bounds for the solutions in
two spatial dimensions. We conclude by illustrating the behaviour of solutions
with numerical experiments in one and two spatial dimensions.
