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 \(n \ge 2\) . 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 \(n \ge 2\) .
