Chemotaxis effect vs logistic damping on boundedness in the 2-D minimal Keller-Segel model

TL;DR

This paper investigates how chemotaxis and logistic damping influence the boundedness of solutions in a 2D Keller-Segel model, introducing a new method to establish explicit bounds and analyze their dependence on model parameters.

Contribution

It presents a novel, simple approach to prove boundedness in the 2D Keller-Segel model with logistic source, providing explicit bounds and parameter dependence analysis.

Findings

Established boundedness of solutions under various parameters.

Derived explicit upper bounds depending on initial data and domain.

First qualitative boundedness result for this model.

Abstract

In this paper, we study chemotaxis effect vs logistic dampening on boundedness for the two-dimensional minimal Keller-Segel model with logistic source in a 2-D smooth and bounded domain. It is well-known that this model allows only for global and uniform-in-time bounded solutions for any chemotactic strength and logistic dampening. Here, we carefully employ a simple and new method to regain its boundedness and, with particular attention to how boundedness depends qualitatively on the coefficient of chemotactic strength and logistic dampening rate. Up to a scaling constant depending only on initial data and the domain, we provide explicit upper bounds for the the solution components of the corresponding initial-boundary value problem. This qualitative boundedness results seems to be the first result in the regard.