On the parabolic-elliptic limit of the doubly parabolic Keller--Segel system modelling chemotaxis

TL;DR

This paper proves strong convergence of solutions from the parabolic-parabolic Keller--Segel system to the parabolic-elliptic model in the plane as a key parameter approaches zero, using advanced space-time estimates.

Contribution

It provides new strong convergence results and global existence conditions for Keller--Segel models, enhancing understanding of chemotaxis system limits.

Findings

Established strong convergence in solutions as parameter tends to zero.

Proved global existence of solutions under smallness assumptions.

Developed space-time estimates for nonintegrable solutions.

Abstract

We establish new convergence results, in strong topologies, for solutions of the parabolic-parabolic Keller--Segel system in the plane, to the corresponding solutions of the parabolic-elliptic model, as a physical parameter goes to zero. Our main tools are suitable space-time estimates, implying the global existence of slowly decaying (in general, nonintegrable) solutions for these models, under a natural smallness assumption.