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Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source

Mario Fuest, Johannes Lankeit, Masaaki Mizukami

TL;DR

The paper analyzes a fully parabolic Keller–Segel-type chemotaxis system with spatially heterogeneous logistic damping in a two-dimensional bounded domain and proves that the blow-up set is contained in the zero set of the damping coefficient, $\\mathcal{B} \\subseteq \\{ x \\in \\overline{\\Omega} : \\mu(x) = 0 \\}$. The authors combine spatially localized maximal Sobolev regularity for the parabolic $v$-equation with carefully designed cutoff functions and a localized energy functional for $u^p$ to obtain local $L^{\\infty}$ control near potential singularities; the analysis proceeds from global estimates to local bounds around points with positive damping. This extends blow-up localization results from the parabolic–elliptic setting to the fully parabolic system and highlights how spatial heterogeneity in the logistic term governs singularity formation. The result provides a rigorous mechanism by which positive logistic damping prevents blow-up away from its zero-set, with potential implications for models of spatially structured populations under resource limitations.

Abstract

We consider the fully parabolic, spatially heterogeneous chemotaxis-growth system \begin{align*} \begin{cases} u_t = Δu - \nabla\cdot(u\nabla v) + κ(x)u-μ(x)u^2, \\ v_t = Δv - v + u \end{cases} \end{align*} in bounded domains $Ω\subset \mathbb{R}^2$ and show that the blow-up set is contained in the set of zeroes of $μ$.

Location of blow-up points in fully parabolic chemotaxis systems with spatially heterogeneous logistic source

TL;DR

The paper analyzes a fully parabolic Keller–Segel-type chemotaxis system with spatially heterogeneous logistic damping in a two-dimensional bounded domain and proves that the blow-up set is contained in the zero set of the damping coefficient, . The authors combine spatially localized maximal Sobolev regularity for the parabolic -equation with carefully designed cutoff functions and a localized energy functional for to obtain local control near potential singularities; the analysis proceeds from global estimates to local bounds around points with positive damping. This extends blow-up localization results from the parabolic–elliptic setting to the fully parabolic system and highlights how spatial heterogeneity in the logistic term governs singularity formation. The result provides a rigorous mechanism by which positive logistic damping prevents blow-up away from its zero-set, with potential implications for models of spatially structured populations under resource limitations.

Abstract

We consider the fully parabolic, spatially heterogeneous chemotaxis-growth system \begin{align*} \begin{cases} u_t = Δu - \nabla\cdot(u\nabla v) + κ(x)u-μ(x)u^2, \\ v_t = Δv - v + u \end{cases} \end{align*} in bounded domains and show that the blow-up set is contained in the set of zeroes of .
Paper Structure (8 sections, 13 theorems, 25 equations)

This paper contains 8 sections, 13 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper.
  3. Preliminaries
  4. Spatially global estimates
  5. Spatially localized estimates
  6. Localized space-time gradient bounds for v
  7. The functional int phi up
  8. Localized L infty bounds for u: proof of Theorem \ref{['mainthm1']}

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^2$ be a smooth, bounded domain, $\kappa,\mu \in C^0(\overline{\Omega})$ with $\mu\ge 0$, $0 \le u_0 \in C^0(\overline{\Omega})$, $0 \le v_0 \in W^{1, \infty}(\Omega)$, $T_{\mathrm{max}} \in (0, \infty]$ and let $(u, v) \in \left( C^{0}(\overline{\Omega} \times [0, T_{\m

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • ...and 4 more