  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.
