This paper is concerned with the Keller-Segel system with flux limitation and nonlinear signal production \(\begin{aligned} \begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{u_t} = \Delta u - \nabla \cdot \left( {uf(|\nabla v|^2)\nabla v} \right) ,}& \\ {0 = \Delta v -\mu (t)+u^{\kappa },\quad \quad \mu (t):=\frac{1}{|\Omega |}\int _{\Omega } u^{\kappa }(x,t)dx}& \end{array}} \right. \end{aligned} \end{aligned}\) in a smooth bounded domain \(\Omega \subset {\mathbb {R}^n}\) \((n\ge 2)\) with no-flux boundary conditions, where \(\kappa \in (0,1]\) and the function \(f\in C^2([0,\infty ))\) fulfills \(\begin{aligned} f(\xi )=(1+\xi )^{-\alpha },\quad \text {for all }\xi \ge 0 \end{aligned}\) with \(\alpha \in \mathbb {R}\) . If either \(\alpha \in \mathbb {R}\) , \(0<\kappa \le \frac{1}{n}\) or \(\alpha >\frac{n\kappa -2}{2(n\kappa -1)}\) , \(\frac{1}{n}<\kappa \le 1\) , then for suitably regular initial data, the solution of the corresponding initial-boundary value problem globally exists and is globally bounded. However, if \(\frac{2}{n}<\kappa \le 1\) and \(0\le \alpha <\frac{n\kappa -2}{2(n\kappa -1)}\) , this system possesses radially symmetric solutions blowing up in finite time, confirming that the number \(\alpha =\frac{n\kappa -2}{2(n\kappa -1)}\) is critical in differentiating global existence from possible blow-up in the case \(\alpha \ge 0\) .
