In this paper, we consider the Cauchy problem of the following parabolic–elliptic–ODE system with indirect signal production mechanism \( \left\{ \begin{aligned} u^*_t&= \Delta u^*- \nabla \cdot \big (u^*\nabla v^*\big ),&x\in {\mathbb {R}}^2, \, t>0,\\ 0&= \Delta v^*+ w^*,&x\in {\mathbb {R}}^2, \, t>0,\\ \tau ^*w^*_t&= -\delta w^*+ u^*,&x\in {\mathbb {R}}^2, \, t>0, \\ u^*(x, 0)&= u^*_0(x), \quad w^*(x,0) = w^*_0(x),&x\in {\mathbb {R}}^2 \end{aligned} \right. \) with \(\tau ^*>0\) and \(\delta >0\) . Our first result asserts that for all reasonably regular initial data, this system admits a unique global solution, which drastically differs from the classical Keller–Segel system with direct signal production mechanism obtained by taking \(\tau ^*=0\) formally. Moreover, the solution remains uniformly bounded whenever the mass \(\int _{{\mathbb {R}}^2}u^*_0(x)\textrm{d}x<4\pi \delta \) . Our second result further reveals that in the radially symmetric setting, there is a threshold value \(\int _{{\mathbb {R}}^2}u^*_0(x)\textrm{d}x=8\pi \delta \) separating two different behaviors: all global solutions are uniformly bounded when the mass is below \(8\pi \delta \) , while there are unbounded solutions starting from initial conditions having a mass exceeding \(8\pi \delta \) .
