Finite-time blowup in a parabolic-parabolic-elliptic chemotaxis model involving indirect signal production

This paper is concerned with a three-component chemotaxis model accounting for indirect signal production,reading as $u_t=\mathrm{\nabla }\cdot (\mathrm{\nabla }u-u\mathrm{\nabla }v)$,$v_t=\mathrm{\Delta }v-v+w$ and $0=\mathrm{\Delta }w-w+u$,posed in a ball of ${\mathbb{R}}^n$ with $n\ge 5$,subject to homogeneous Neumann boundary conditions.The system is a Nagai-type variant of its fully parabolic version that has a four-dimensional critical mass phenomenon concerning blowup in finite or infinite time according to the seminal works of Fujie and Senba [J. Differential Equations, 263 (2017), 88–148; 266 (2019), 942–976].We prove that for any prescribed mass $m>0$, there exist radially symmetric and positive initial data $(u_0,v_0)\in C^0(\bar{\mathrm{\Omega }})\times C^2(\bar{\mathrm{\Omega }})$ with ${\int }_{\mathrm{\Omega }}{u}_0=m$ such that the corresponding solutions blow up in finite time.