Large global solutions of the parabolic-parabolic Keller-Segel system in higher dimensions

TL;DR

This paper proves that large initial data lead to global solutions of the Keller-Segel system in higher dimensions when the chemoattractant diffusion parameter is sufficiently large, extending previous two-dimensional results.

Contribution

It extends the understanding of global existence for the Keller-Segel system to higher dimensions and improves size conditions on initial data, showing near-optimal results for large diffusion parameters.

Findings

Global solutions exist for arbitrary initial data with large diffusion parameter $ au$ in dimensions $d \,\ge\, 3$.

Size conditions on initial data are nearly optimal, up to a logarithmic factor in $ au$.

Toy models demonstrate finite-time blowup for certain large solutions, complementing the global existence results.

Abstract

We study the global existence of the parabolic-parabolic Keller-Segel system in $\R^d , d \ge 2$. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter $\tau$ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler, Guerra \& Karch (2015) and Corrias, Escobedo \& Matos (2014). Our analysis improves earlier results and extends them to any dimension $d \ge 3$. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in $\tau$, when $\tau>>1$: we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller-Segel system. For these toy models, we establish in a companion paper [4] finite time blowup for a class of large solutions.