This paper investigates the repulsion–consumption system under no-flux/Dirichlet conditions for u and v in a ball \(B_R(0) \subset \mathbb {R}^n \) . When \(\tau \in \{0,1\}\) , \(n\geqslant 2\) and \(0<S(u)\leqslant K(1+u)^{\beta }\) for \(u \geqslant 0\) with \(\beta \in (0,\frac{n+2}{2n})\) and \(K>0\) , we show that for any given radially symmetric initial data, the problem ( \(\star \) ) possesses a global bounded classical solution. Conversely, when \(\tau =0\) , \(n=2\) and \(S(u) \geqslant k u^{\beta }\) for \(u \geqslant 0\) with \(\beta >1\) and \(k>0\) , for any given radially symmetric initial data \(u_0\) , there exists a constant \(M^{\star }=M^{\star }\left( u_0\right) >0\) with the property that whenever the boundary signal level \(M\geqslant M^{\star }\) , the corresponding solution blows up in finite time. Our results imply that \(\beta = 1\) is optimal when \(\tau = 0\) and \(n = 2\) .
