Nonlocal nonlinear reaction preventing blow-up in Keller-Segel system

TL;DR

This paper demonstrates that a nonlocal nonlinear reaction term in the Keller-Segel chemotaxis model can prevent blow-up, ensuring global solutions under certain conditions through advanced a priori estimate techniques.

Contribution

It introduces a novel nonlocal nonlinear reaction mechanism that guarantees global existence of solutions in chemotaxis models, extending previous blow-up prevention methods.

Findings

Nonlocal nonlinear reaction prevents blow-up in chemotaxis system.

Global classical solutions exist for certain exponents.

Modified Moser-Alikakos iteration is effective for a priori estimates.

Abstract

This paper is devoted to the analysis of non-negative solutions for the chemotaxis model with nonlocal source in bounded domain. The qualitative behavior of solutions is determined by the nonlinearity from the aggregation and the reaction. The nonlocal nonlinear Fisher-KPP reaction helps preventing blow-up phenomena in chemotaxis system. For appropriately chosen exponents, the global existence of classical solutions is proved with arbitrary initial data, where a modified Moser-Alikakos iteration method plays a key role in the a priori estimates.