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

Xuan Mao, Meng Liu, Yuxiang Li

TL;DR

The paper addresses finite-time blowup for a fully parabolic chemotaxis system with indirect signal production in a high-dimensional ball, proving that for any mass \(m>0\) there exist radial, nonnegative initial data with \(\int_\Omega u_0=m\) that blow up in finite time. The authors develop a memory-aware strategy by constructing a Lyapunov-based integral inequality for the cross-term \(\int_\Omega uv\), leveraging Pohožaev-type testing and delicate spatial-temporal estimates to bound the Laplacian of the concentration \(\Delta v\) and the associated energy, despite the presence of the memory term \(w_t\). A key contribution is showing that low initial energy forces blowup, and that initial data with arbitrarily large negative energy can be generated, implying a dense blowup regime in the admissible set. Together, these results advance understanding of blowup phenomena in high-dimensional chemotaxis with indirect signaling and memory effects.

Abstract

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 = Δu - \nabla\cdot(u\nabla v),\\ v_t = Δv - v + w,\\ w_t = Δw - 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 $(u_0,v_0,w_0)\in C^0(\overlineΩ)\times C^2(\overlineΩ)\times C^2(\overlineΩ)$ with $\int_Ωu_0 = m$ such that the corresponding classical solutions blow up in finite time. The key ingredient is a novel integral inequality for the cross-term integral $\int_Ωuv$ constructed via a Lyapunov functional.

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

TL;DR

The paper addresses finite-time blowup for a fully parabolic chemotaxis system with indirect signal production in a high-dimensional ball, proving that for any mass there exist radial, nonnegative initial data with that blow up in finite time. The authors develop a memory-aware strategy by constructing a Lyapunov-based integral inequality for the cross-term , leveraging Pohožaev-type testing and delicate spatial-temporal estimates to bound the Laplacian of the concentration and the associated energy, despite the presence of the memory term . A key contribution is showing that low initial energy forces blowup, and that initial data with arbitrarily large negative energy can be generated, implying a dense blowup regime in the admissible set. Together, these results advance understanding of blowup phenomena in high-dimensional chemotaxis with indirect signaling and memory effects.

Abstract

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 = Δu - \nabla\cdot(u\nabla v),\\ v_t = Δv - v + w,\\ w_t = Δw - w + u, \end{cases} \end{equation*} posed on a ball in with , 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 , there exist radially symmetric and nonnegative initial data with such that the corresponding classical solutions blow up in finite time. The key ingredient is a novel integral inequality for the cross-term integral constructed via a Lyapunov functional.

Paper Structure

This paper contains 10 sections, 14 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries. Energy functional and basic estimates
  3. Construction of an integral inequality
  4. Notations and setup
  5. Interpolation inequalities for concentration v
  6. Spatial-temporal estimates of uv
  7. Spatial-temporal estimates of Delta v
  8. Refined spatial-temporal estimates of uv
  9. Finite-time blowup enforced by low initial energy
  10. Initial data with large negative energy

Key Result

Theorem 1.1

Assume $\Omega = B_R \subset\mathbb R^n$ for some $n\geq5$ and $R>0$. Let $\tilde{m} > 0$ and $A > 0$ be given. Then there exist positive constants $K(\tilde{m},A)$ and $T(\tilde{m},A)$ such that if $(u_0,v_0,w_0)$ from the set satisfies $\mathcal{F}_{1,1}(u_0,v_0,w_0) < - K(\tilde{m},A)$, then the corresponding classical solution given by Proposition prop: local existence and uniqueness blows up

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • ...and 17 more