Sufficient conditions for global boundedness in inertial chemotaxis systems with logistic growth

This paper investigates a chemotaxis model described by the following system of partial differential equations:

$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \Delta u - \nabla \cdot (u \mathbf{w}) + \gamma(u - u^{\alpha}), & x \in \Omega, \, t > 0, \\ v_{t} = \Delta v - v + u, & x \in \Omega, \, t > 0, \\ \mathbf{w}_{t} = \Delta \mathbf{w} - \mathbf{w} + \chi \nabla v, & x \in \Omega, \, t > 0, \\ \partial_{\mathbf{n}} u = \partial_{\mathbf{n}} v = 0, \quad \mathbf{w} = 0, & x \in \partial \Omega, \end{array} \right. \end{eqnarray*} $

posed on a smooth, bounded domain $ \Omega \subset \mathbb{R}^n $, where $ \chi > 0 $, $ \gamma \geq 0 $, and $ \alpha > 1 $ are parameters. By employing $ L^p $-estimates and a carefully constructed bootstrap iteration method, we analyze the intricate mathematical relationships governing the system. Our findings demonstrate the existence of globally bounded solutions for $ n \geq 4 $, contingent upon the conditions that $ \gamma > 0 $ and $ \alpha > \frac{3n + 6}{n + 8} $. These findings significantly refine the known parameter constraints for global solution existence. In particular, our work improves upon previous studies—such as those by Zhang et al. (2019), Dong and Peng (2021), and Mu and Tao (2022)—which typically imposed the stricter condition $ \alpha > \frac{1}{2} + \frac{n}{4} $. Our analysis thus provides sharper criteria for $ \alpha $, broadening the understanding of solution behavior in chemotaxis systems.