  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.
