  We prove the existence of global-in-time weak solutions to a version of the
parabolic-parabolic Keller-Segel system in one spatial dimension. If the
coupling of the system is suitably weak, we prove convergence of those
solutions to the unique equilibrium with an exponential rate. Our proofs are
based on an underlying gradient flow structure with respect to a mixed
Wasserstein-$L^2$ distance.

Analysis of PDEs