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 $\varepsilon\ge0$. The nonlinearities and potentials are chosen such that in the decoupled system for $\varepsilon=0$, the evolution is metrically contractive, with a global rate $\Lambda>0$. 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 $\Lambda_\varepsilon=\Lambda-K\varepsilon$. The proof combines results from the theory of metric gradient flows with further variational methods and functional inequalities.