Uniform boundedness for the two-dimensional Keller-Segel system with Gompertz growth

TL;DR

This paper investigates the impact of a Gompertz growth term on the two-dimensional Keller-Segel system, demonstrating conditions under which solutions remain globally bounded despite the weaker damping effect.

Contribution

It introduces a novel analysis of the Keller-Segel model with Gompertz growth, showing boundedness results under minimal damping conditions.

Findings

Solutions exist globally and are bounded with Gompertz growth

Weaker damping mechanisms can still prevent cell aggregation

Conditions for boundedness are explicitly identified

Abstract

It is known that in two dimensions the classical Keller-Segel model can lead to cell aggregation. This behavior can be controlled by adding a logistic growth term with quadratic decay. Researchers have tried to find weaker damping mechanisms that can still stabilize the system. Previous work showed that, under suitable assumptions on the initial cell distribution, even weaker growth terms than the classical logistic one can prevent aggregation. In this paper, we study the effect of a Gompertz-type growth term in a minimal two-dimensional chemotaxis model. This term provides a weaker damping effect than those previously considered. We analyze how it influences the system and identify conditions that guarantee that solutions exist for all time and remain bounded.