Finite-time blowup in a parabolic-parabolic-elliptic chemotaxis model involving indirect signal production
Xuan Mao, Yuxiang Li
TL;DR
The paper addresses finite-time blowup in a Nagai-type parabolic-parabolic-elliptic chemotaxis model with indirect signal production in dimensions $n\ge5$. It develops an energy-dissipation framework with a Lyapunov functional $\mathcal{F}$ and dissipation $\mathcal{D}$ and proves a bound $-\mathcal{F} \le C(\mathcal{D}^{\theta}+1)$ for some $\theta\in(0,1)$, leading to a differential inequality that enforces blowup for low-energy initial data. It further shows that for any mass $m>0$ there exist radially symmetric initial data causing finite-time blowup, and that low-energy data are dense in the admissible set via concentrated perturbations that drive $\mathcal{F}$ to $-\infty$, establishing the robustness and ubiquity of the blowup mechanism. These results illuminate how indirect signal production and high-dimensional diffusion-chemotaxis interactions can still yield finite-time singularities, enriching the theory of chemotaxis models with indirect signaling.
Abstract
This paper is concerned with a three-component chemotaxis model accounting for indirect signal production,reading as $u_t = \nabla\cdot(\nabla u - u\nabla v)$,$v_t = Δv - v + w$ and $0 = Δw - w + u$,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 $(u_0,v_0)\in C^0(\overlineΩ)\times C^2(\overlineΩ)$ with $\int_Ωu_0 = m$ such that the corresponding solutions blow up in finite time.