In this talk, we investigate the fully parabolic Keller-Segel system with a nonlinear sensitivity of the form
\begin{equation}
\left{
\begin{aligned}
&\partial_{t}u=\Delta{u}-\nabla\cdot\big(u(1+u)^{-\alpha}\nabla{v}\big), &&(x,t)\in\Omega\times(0,\infty),
&\partial_{t}v=\Delta{v}-v+u, &&(x,t)\in\Omega\times(0,\infty)
\end{aligned}
\right.
\end{equation}
in a bounded domain with smooth boundary $\Omega\subset\mathbb{R}^N(N\geq3)$. We prove that for all suitably regular initial data, an associated no-flux-Dirichlet initial-boundary value problem possesses a globally bounded classical solution if $\alpha>1-\frac{2}{N}$. On the other hand, we construct finite-time blow-up solutions for the case of $0<\alpha<1-\frac{2}{N}$ in the radially symmetric setting. Our result indicates that $\alpha_c:=1-\frac{2}{N}$ is a critical blow-up exponent. This is a joint work with Dr Ying Dong and Professor Yulan Wang.