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

Trend to Equilibrium for Systems with Small Cross-Diffusion