A note on the variational analysis of the parabolic-parabolic Keller-Segel system in one spatial dimension

TL;DR

This paper establishes the existence of global weak solutions for a one-dimensional parabolic-parabolic Keller-Segel system and demonstrates their exponential convergence to equilibrium under weak coupling, using a gradient flow approach.

Contribution

It introduces a novel analysis of the Keller-Segel system in one dimension, proving existence and convergence results via a gradient flow framework.

Findings

Global weak solutions exist in one dimension.

Solutions converge exponentially to equilibrium under weak coupling.

Gradient flow structure is key to the analysis.

Abstract

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.