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

Abstract

It is well known that the Neumann initial-boundary value problem for the minimal Keller-Segel chemotaxis system in a 2D bounded smooth domain has no blow-ups for any presence of logistic source of cell kinetics. Here, for a large class of cell kinetics including sub-logistic sources, we find an explicit condition involving the chemotactic strength, the asymptotic “damping” rate, and the initial mass of cells to ensure the uniform-in-time boundedness for the corresponding 2D Neumann initial-boundary value problem. Our finding in particular shows that sub-logistic source can prevent blow-up in 2D, indicating that logistic damping is not the weakest damping to guarantee boundedness for the 2D Keller-Segel minimal chemotaxis model.