Global existence of solutions to the fully parabolic chemotaxis system with logistic source under nonlinear Neumann boundary conditions

TL;DR

This paper proves the existence of globally bounded solutions for a chemotaxis system with logistic growth and nonlinear boundary conditions in any spatial dimension, extending previous results limited to specific dimensions.

Contribution

It generalizes the conditions for global bounded solutions in chemotaxis models with nonlinear boundary conditions to all dimensions $n \, \geq \, 2$, for $p<\frac{3}{2}$.

Findings

Global bounded solutions exist for all dimensions $n \geq 2$ when $p<\frac{3}{2}$.

Extends previous dimension-specific results to a general setting.

Provides theoretical foundation for chemotaxis models with nonlinear boundary conditions.

Abstract

We study the existence of global boundedness solutions to the fully parabolic chemotaxis systems with logistic sources, $ru- \mu u^2$, under nonlinear Neumann boundary conditions, $\frac{\partial u}{\partial \nu }= |u|^p$ where $p >1 $ in smooth bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$. A recent study by Le (2023) has shown that the logistic sources can ensure that solutions are global and bounded when $n =2$ with $p < \frac{3}{2}$ and $n=3$ with $p <\frac{7}{5}$. In this paper, we extend the previous findings by demonstrating the existence of global bounded solutions when $p< \frac{3}{2}$ in any spatial dimension $n \geq 2$.