This paper deals with a repulsive chemotaxis–consumption system \(\begin{aligned} \left\{ \begin{aligned} u_t&=\nabla \cdot (D(u)\nabla u)+ \nabla \cdot (S(u) \nabla v),\\ 0&=\Delta v-uv, \end{aligned} \right. \end{aligned}\) is considered along with the boundary conditions \((D(u)\nabla u+S(u)\nabla v)\cdot \nu |_{\partial \Omega }=0\) and \(v|_{\partial \Omega }=M\) , where \(\Omega =B_R(0)\subset {\mathbb {R}}^n\) is a smoothly bounded domain. When \(n=2\) , \(0<D(u)\le K_D(1+u)^{-\alpha }\) and \(S(u)\ge K_Su^{\beta }\) for all \(u>0\) with some \(K_D>0\) , \(K_S>0\) , \(\alpha >0\) and \(\beta >1\) , then for any nonnegative initial data \(u_0\) , one can find \(M_\star \left( u_0\right) >0\) with the boundary signal level \(M\ge M_\star (u_0)\) , the corresponding radially symmetric solution blowsup in finite time. Conversely, when \(n\ge 2\) , \(D(u)\ge k_D(1+u)^{-\alpha }\) and \(0<S(u)\le k_S(1+u)^{\beta }\) with \(k_D>0\) , \(k_S>0\) , \(\alpha \le 0\) , \(\beta >0\) and \(\alpha +\beta <\frac{n+2}{2n}\) , it is shown that for any nonnegative radially symmetric initial data, the system possesses a global bounded classical solution.
