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

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