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

TL;DR

This paper demonstrates that in a three-component chemotaxis model with indirect signal production, solutions can blow up in finite time for any initial mass, extending understanding of blowup phenomena in such systems.

Contribution

It proves finite-time blowup for any initial mass in a chemotaxis model with indirect signal production, a significant extension of previous results on related models.

Findings

Finite-time blowup occurs for any initial mass.

Radially symmetric positive initial data can lead to blowup.

The model extends previous Nagai-type chemotaxis systems.

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 in a ball of $\mathbb R^n$ with $n\geq5$,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.