This paper is concerned with a critical mass phenomenon in the following nonlinear Keller–Segel system with indirect signal production proposed on \(\Omega :=B_R(0) \subset \mathbb {R}^n(n\geqslant 2) \) and subjected to homogeneous Neumann boundary conditions. According to the paper (Jin and Li in Nonlinear Anal 89, Paper No. 104523, 2026), the system ( \(\star \) ) in radial setting admits \(\frac{2}{n}\) as a critical exponent in the sense that if \(\alpha < \frac{2}{n}\) , all solutions exist globally and if \(\alpha > \frac{2}{n}\) , there exist finite-time blowup solutions. In this paper, we further demonstrate a critical mass phenomenon in the supercritical case \( \alpha \geqslant \frac{2}{n}\) . More precisely, there exists a critical mass \(m_c:= m_c(n, R, \alpha )\) such that for arbitrary nonconstant radial initial data \((u_0, w_0)\) satisfying \(\int _{\Omega } u_0>m_c\) , and for all \(r \in (0,R)\) , if \(\alpha >\frac{2}{n}\) , the corresponding solution blows up in finite time; if \(\alpha =\frac{2}{n}\) , the corresponding solution blows up in infinite time and forms a Dirac-type distribution as times goes to infinity.
there exist nonconstant, regular, and radial initial data \((u_0, w_0)\) with \(\int _{\Omega } u_0<m_c\) such that the corresponding solution is global.
