Global existence of solutions to the chemotaxis system with logistic source under nonlinear Neumann boundary condition

TL;DR

This paper proves the global existence and boundedness of solutions to a chemotaxis system with logistic source under nonlinear Neumann boundary conditions, depending on parameters like p, n, and μ.

Contribution

It establishes new conditions under which classical solutions to the chemotaxis system exist globally and remain bounded, extending previous results to nonlinear boundary conditions.

Findings

Global solutions exist for p<3/2 when n=2 or μ is large when n≥3.

Solutions are unique, positive, and bounded in the domain over time.

Results apply to both parabolic-elliptic and parabolic-parabolic chemotaxis systems.

Abstract

We consider classical solutions to the chemotaxis system with logistic source $f(u) := au-\mu u^2$ under nonlinear Neumann boundary condition $\frac{\partial u}{ \partial \nu } = |u|^{p}$ with $p>1$ in a smooth convex bounded domain $\Omega \subset \mathbb{R}^n$ where $n \geq 2$. This paper aims to show that if $p<\frac{3}{2}$, and $\mu >0$, $n=2$, or $\mu$ is sufficiently large when $n\geq 3$, then the parabolic-elliptic chemotaxis system admits a unique positive global-in-time classical solution that is bounded in $\Omega \times (0, \infty)$. The similar result is also true if $p<\frac{3}{2}$, $n=2$, and $\mu>0$ or $p<\frac{7}{5}$, $n=3$, and $\mu $ is sufficiently large for the parabolic-parabolic chemotaxis system.