Finite-time blowup in a five- and higher-dimensional chemotaxis model with indirect signal production

Abstract: 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 on a ball in $\mathbb R^n$ with $n\geq5$, subject to homogeneous Neumann boundary conditions. It is suggested that the system has a four-dimensional critical mass phenomenon concerning blowup in finite or infinite time according to the work of Fujie and Senba [J. Differential Equations, 263 (2017), 88--148; 266 (2019), 942--976]. In this paper, 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.