A critical nonlinearity for blow-up in a higher-dimensional chemotaxis system with indirect signal production

The Neumann problem in balls $ \Omega\subset\mathbb{R}^n $, $ n\in\{3,4\} $, for the chemotaxis system$ \left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u\nabla v), \\ 0 = \Delta v - \mu^{(w)}(t) + w, \qquad \mu^{(w)}(t) = \frac{1}{|\Omega|}\displaystyle {\int }_\Omega w,\\ w_t = \Delta w - w + f(u), \end{array} \right. $is considered. Under the assumption that $ f\in C^1([0,\infty)) $ is such that $ f(\xi) \geq k\xi^\sigma $ for all $ \xi\geq 0 $ and some $ k>0 $ and $ \sigma>\frac{4}{n} $, it is shown that finite-time blow-up occurs for some radially symmetric solutions.