Nonlocal reaction preventing blow-up in the supercritical case of chemotaxis

TL;DR

This paper investigates how nonlocal nonlinear sources in the Keller-Segel chemotaxis model can prevent blow-up and ensure global bounded solutions, especially in supercritical regimes, by balancing aggregation and damping effects.

Contribution

It demonstrates that nonlocal nonlinear terms can prevent chemotactic collapse and establish global existence of solutions in supercritical cases, extending understanding of chemotaxis models.

Findings

Nonlocal nonlinear sources prevent blow-up in supercritical chemotaxis models.

Global classical solutions exist under certain exponent conditions.

Solutions converge to a constant equilibrium state.

Abstract

This paper studies the non-negative solutions of the Keller-Segel model with a nonlocal nonlinear source in a bounded domain. The competition between the aggregation and the nonlocal reaction term is highlighted: when the growth factor is stronger than the dampening effect, with the help of the nonlocal term, the model admits a classical solution which is uniformly bounded. Moreover, when the growth factor is of the same order compared to the dampening effect, the nonlocal nonlinear exponents can prevent the chemotactic collapse. Global existence of classical solutions is shown for an appropriate range of the exponents as well as convergence to the constant equilibrium state.