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

This paper is concerned with a parabolic-parabolic-parabolic chemotaxis system with indirect signal production, modelling the impact of phenotypic heterogeneity on population aggregation \begin{equation*} \begin{cases} u_t = Δu - \nabla\cdot(u\nabla v),\ v_t = Δv - v + w,\ w_t = Δw - w + u, \end{cases} \end{equation*} posed on a ball in ${\mathbb{R}}^n$ with $n\ge 5$, subject to homogeneous Neumann boundary conditions. The system 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 nonnegative initial data $(u_0,v_0,w_0)\in C^0(\bar{\mathrm{\Omega }})\times C^2(\bar{\mathrm{\Omega }})\times C^2(\bar{\mathrm{\Omega }})$ with ${\int }_{\mathrm{\Omega }}{u}_0=m$ such that the corresponding classical solutions blow up in finite time. The key ingredient is a novel integral inequality for the cross-term integral ${\int }_{\mathrm{\Omega }}uv$ constructed via a Lyapunov functional.