Sharp well-posedness and blowup results for parabolic systems of the Keller-Segel type

TL;DR

This paper investigates modified Keller-Segel models, establishing conditions for global existence or finite-time blowup based on initial data size relative to the diffusion parameter, highlighting the sharpness of these conditions.

Contribution

It introduces toy models of Keller-Segel systems and determines sharp size conditions for global solutions versus blowup, refining understanding of solution behavior.

Findings

Global solutions for small initial data relative to diffusion parameter

Finite time blowup for larger initial data

Sharpness of size condition for large diffusion parameter

Abstract

We study two toy models obtained after a slight modification of the nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy models, both consisting of a system of two parabolic equations, we establish that for data which are, in a suitable sense, smaller than the diffusion parameter $\tau$ in the equation for the chemoattractant, we obtain global solutions, and for some data larger than $\tau$ , a finite time blowup. In this way, we check that our size condition for the global existence is sharp for large $\tau$ , up to a logarithmic factor.