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 = \Delta u - \nabla\cdot(u\nabla v),\\ v_t = \Delta v - v + w,\\ w_t = \Delta w - w + u, \end{cases} \end{equation*} posed on a ball in $\mathbb R^n$ with $n\geq5$, 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 $(u_0,v_0,w_0)\in C^0(\overline{\Omega})\times C^2(\overline{\Omega})\times C^2(\overline{\Omega})$ with $\int_\Omega u_0 = m$ such that the corresponding classical solutions blow up in finite time. The key ingredient is a novel integral inequality for the cross-term integral $\int_\Omega uv$ constructed via a Lyapunov functional.