  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 $n \geq 2$.
