We consider the following variant of the Keller-Segel system for chemotaxis,
\begin{align*}
\partial_t\rho &= \Delta(\rho^2) + \nabla\cdot\big(\rho[W+\varepsilon \phi(c)]\big), \\
\partial_t c &= \Delta c - \kappa c - \varepsilon \rho\phi'(c),
\end{align*}
with nonlinear mobility of the bacteria and a (possibly non-linear) chemotactic sensitivity \(\phi\).
Solutions \((\rho,c)\) to this system of equations constitute a gradient flow in the coupled Wasserstein-\(L^2\)-metric.
The flow's driving functional has no useful convexity properties with respect to that metric,
but it is the \(\varepsilon\)-perturbation of an entropy functional that is \(\lambda\)-convex,
with a positive \(\lambda\) if the external potential \(W\) is sufficiently confining.
With the help of this special structure, we prove the global-in-time existence of weak solutions,
and their convergence to equilibrium, exponentially fast in time.
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