Sub-logistic source can prevent blow-up in the 2D minimal Keller-Segel chemotaxis system

TL;DR

This paper demonstrates that sub-logistic sources in the 2D Keller-Segel chemotaxis system can prevent blow-up, revealing a new damping mechanism that ensures boundedness beyond traditional logistic terms.

Contribution

It introduces the concept that sub-logistic sources can prevent blow-up in 2D chemotaxis models, expanding understanding of damping effects.

Findings

Sub-logistic sources prevent blow-up in 2D chemotaxis systems.

Logistic damping is not the minimal requirement for boundedness.

New damping phenomena identified in chemotaxis models.

Abstract

It is well-known that the Neumann initial-boundary value problem for the minimal-chemotaxis-logistic system in a 2D bounded smooth domain has no blow-up for any choice of parameters. Here, for a large class of kinetic terms including sub-logistic sources, we show that the corresponding 2D Neumann initial-boundary value problems do not possess any blow-up. This illustrates a new phenomenon that even a class of sub-logistic sources can prevent blow-up for the 2D problem, indicating that logistic damping is not the weakest damping to guarantee uniform-in-time boundedness for the 2D minimal Keller-Segel chemotaxis model.