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

The Neumann problem in balls $\mathrm{\Omega }\subset {\mathbb{R}}^n$, $n\in \{3,4\}$, for the chemotaxis system \begin{equation*} \left{ \begin{array}{ll} u_t = Δu - \nabla \cdot (u\nabla v), \[1mm] 0 = Δv - μ^{(w)}(t) + w, \quad μ^{(w)}(t) = \frac{1}{|Ω|}\int_Ωw \[1mm] w_t = Δw - w + f(u), \end{array} \right. \end{equation*} is considered. Under the assumption that $f\in C^1([0,\mathrm{\infty }))$ is such that $f(\xi )\ge k{\xi }^{\sigma }$ for all $\xi \ge 0$ and some $k>0$ and $\sigma >\frac{4}{n}$, it is shown that finite-time blow-up occurs for some radially symmetric solutions.