Chemotaxis models with mixed mechanisms: boundedness in growth-dominated regimes

TL;DR

This paper investigates a chemotaxis-growth system with complex reactions and damping, demonstrating conditions under which solutions are globally bounded, emphasizing the interplay of chemotaxis, damping, and nonlocal effects in preventing blow-up.

Contribution

It introduces a novel two-step approach to prove global boundedness in growth-dominated chemotaxis models with mixed mechanisms.

Findings

Solutions exist globally and are uniformly bounded under certain conditions.

Mass control is achieved through a two-step process, addressing non-automatic mass regulation.

Chemotaxis, damping, and nonlocal effects collaboratively prevent blow-up.

Abstract

We study a chemotaxis-growth system with nonlinear local and nonlocal reactions and gradient-dependent damping. Under suitable conditions on the system parameters and spatial dimension, we prove that solutions exist globally in time and remain uniformly bounded. Unlike classical cases, when local growth dominates, mass control is not automatic. To address this, we use a two-step approach: first ensuring bounded total mass, then establishing full uniform boundedness. The results highlight how chemotaxis, damping, and nonlocal effects interact to prevent blow-up in structured models.