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

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.