Exponential Convergence to Equilibrium for Coupled Systems of Nonlinear Degenerate Drift Diffusion Equations

We study the existence and long-time asymptotics of weak solutions to a system of two nonlinear drift-diffusion equations that has a gradient flow structure in the Wasserstein distance. The two equations are coupled through a cross-diffusion term that is scaled by a parameter \varepsilon\geq 0. The nonlinearities and potentials are chosen such that in the decoupled system for \varepsilon = 0, the evolution is metrically contractive, with a global rate \Lambda > 0\Lambda > 0. The coupling is a singular perturbation in the sense that for any \varepsilon > 0, contractivity of the system is lost. Our main result is that for all sufficiently small \varepsilon > 0, the global attraction to a unique steady state persists, with an exponential rate \Lambda\varepsilon = \Lambda -K\varepsilon for some k > 0. The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities.