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 \(u_t = \nabla \cdot (\nabla u - u\nabla v)\), \(v_t = \Delta v - v + w\) and \(0 = \Delta w - w + u\), 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 \((u_0,v_0)\in C^0(\overline{\Omega })\times C^2(\overline{\Omega })\) with \(\int _\Omega u_0 = m\) such that the corresponding solutions blow up in finite time.