In this work, we investigate the following Keller–Segel system with nonlinear sensitivity and flux limitation \(\begin{aligned} {\left\{ \begin{array}{ll} \begin{array}{lll} u_t=\Delta u-\nabla \cdot \left( \frac{f(u)\nabla v}{(1+|\nabla v|^2)^\frac{\beta }{2}} \right) ,&x \in \Omega , t>0, 0=\Delta v-\mu +u,&x \in \Omega , t>0, \end{array} \end{array}\right. } \end{aligned}\) under homogeneous Neumann-type boundary conditions in a bounded domain \(\Omega \subset \mathbb {R}^n (n \ge 2\) ), where \(\beta \in (0, 1]\) , \( \mu :=\frac{1}{|\Omega |} \int _{\Omega } u_0(x) \,\textrm{d}x \) is positive and \(f\in C^2([0,\infty ))\) with \(f(0)=0\) equals some nonlinear function of the particle density. This paper proves that: Let \(f(u)\ge ku^p\) for all \(u\ge 1\) with some \(k > 0\) and \(p>0\) . If \(\Omega \) is a ball and \(p>(1-\frac{1}{n})\beta +\frac{2}{n}\) , there exist some radially symmetric initial data such that the corresponding solution blows up in finite time.
Let \(f(u)\le K(u+1)^p\) for all \(u\ge 0\) with some \(K > 0\) and \(p>0\) . If \(0<p<(1-\frac{1}{n})\beta +\frac{2}{n}\) , all solutions are global and uniformly bounded.
