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Finite time blowup for Keller-Segel equation with logistic damping in three dimensions

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Pith's one-line read Finite-time blowup solutions exist for the 3D Keller-Segel equation with logistic damping for every damping coefficient below 1/3, matching the known global-existence threshold.

arxiv 2504.12231 v2 pith:NV2UA3BY submitted 2025-04-16 math.AP

Jiaqi Liu , Yixuan Wang , Tao Zhou This is my paper · ORCID
classification math.AP
keywords blowupequationkeller-segeldampinglogisticsolutionapproximateclassical
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A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Keller-Segel equations model how bacteria aggregate by following a chemical they produce. In three dimensions, even a tiny initial mass can lead to a singularity: the density becomes infinite in finite time. Adding a quadratic death term, often called logistic damping, with coefficient μ, can suppress this. Prior work showed that if μ is at least 1/3, every smooth solution stays smooth forever. This paper proves the converse: for any μ below 1/3, there exist smooth solutions with compactly supported, nonnegative initial data that blow up in finite time.

The construction works in two steps. First, the authors ignore diffusion and build a self-similar blowup profile for the resulting aggregation equation. The profile is obtained by solving a second-order ODE with a phase-portrait analysis; it decays as a power law. Because the profile is chosen with a subcritical scaling exponent β < 1/2, the neglected diffusion term is actually a small perturbation in the rescaled variables. Second, the authors prove that this approximate solution is nonlinearly stable: small errors remain small and decay exponentially in the rescaled time. This stability proof uses a specially chosen singular weight and modulation ODEs to control the finite-dimensional unstable directions.

A byproduct is a new countable family of blowup profiles for the classical Keller-Segel equation in three dimensions, obtained by setting μ = 0. These profiles have a different spatial scaling from the previously known self-similar solutions.

Extended reading notes

Core claim

Theorem 1.1: For any 0 ≤ μ < 1/3, there exists a radially nonnegative ρ0 ∈ C∞_0(R3) such that the smooth solution to (KS-D) blows up at finite time T, with ρ(t,x) = 1/(T-t) ( Q(x/(T-t)^β) + ε(t, x/(T-t)^β) ), where Q solves the profile equation (1.8), β = 1/(3(1-μ)) + 1/(2j0) < 1/2, and ||ε(t)||_{H^s} ≤ C (T-t)^{ε̄}. If correct, this gives a complete sharp threshold for logistic damping and a new family of type I blowup profiles for the classical 3D Keller-Segel equation.

Load-bearing premise

The proof rests on the coercivity estimate of Proposition 4.1: there exists a singular weight w = |y|^{-A} + B, with A ≫ 4j0 + 3 and A/4 ∈ Z>0, such that the linearized operator L around the profile Q satisfies (Lg,g)_{L²_w} ≤ -1/8 ||g||²_{L²_w} for all radial g. The proof chooses A and B non-explicitly and relies on the monotonicity and decay properties of Q from Lemma 3.2. If this coercivity fails, the bootstrap in Section 5 cannot control the stable part ε_s, and the nonlinear stability argument collapses.

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Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The theorem rests on standard ODE/PDE tools and on the construction-specific choices of the weight and modulation parameters. No new physical entities are postulated.

free parameters (5)
  • j0 = integer ≥ J = 3(1-μ)/(1-3μ) + 1
    Integer parameter determining the profile's vanishing order and the blowup scale β; chosen freely in the theorem.
  • A = large integer, multiple of 4, with A > 4j0 + 3
    Exponent of the singular weight w = |y|^{-A} + B; chosen sufficiently large to make the linearized operator coercive in Proposition 4.1.
  • B = small positive constant
    Offset of the weight; chosen small enough to absorb lower-order terms in the coercivity estimate.
  • Q_{j0} = -1
    First nonzero Taylor coefficient of the profile, normalized to -1; any negative value works, the amplitude is a normalization.
  • λ0, R2 = λ0 small, R2 large
    Parameters of the initial data; chosen so that the cutoff profile is small in the relevant norms and the initial data is compactly supported.
assumptions (5)
  • standard math Cauchy-Lipschitz theory for the radial ODE system (3.3).
    Used to extend the local solution and to rule out exit scenarios in Lemma 3.2.
  • standard math Brouwer fixed point theorem and Gagliardo-Nirenberg inequalities.
    Used to select initial modulation parameters and to close the bootstrap estimates.
  • domain assumption Local well-posedness of (KS-D) in H^s for s ≥ 2K+6.
    Invoked in the proof of Proposition 5.2 and Theorem 1.1, citing [52] and [44, Appendix A].
  • domain assumption Radial symmetry and nonnegativity of initial data are preserved.
    The construction is restricted to radial solutions; nonnegativity is asserted via [44, Theorem A.1].
  • ad hoc to paper Diffusion term is subcritical in the self-similar variables (λ^{2-4β} ΔΨ is a perturbation).
    This is the central structural assumption that motivates the approximate solution; it is justified a posteriori by the stability analysis.

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Pith. "Pith review of Finite time blowup for Keller-Segel equation with logistic damping in three dimensions." pith.science (2026). https://pith.science/paper/NV2UA3BY

@misc{pith2026250412231,
  author       = {Pith},
  title        = {Pith review of: Finite time blowup for Keller-Segel equation with logistic damping in three dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NV2UA3BY}},
  note         = {Machine review of arXiv:2504.12231}
}
abstract

The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term $-\mu \rho^2$ (modeling density-dependent mortality rate) and show the existence of finite-time blowup solutions with nonnegative density and finite mass for any $\mu \in \big[0,\frac{1}{3}\big)$. This range of $\mu$ is sharp; for $\mu \ge \frac{1}{3}$, the logistic damping effect suppresses the blowup as shown in [Kang-Stevens, 2016] and [Tello-Winkler, 2007]. A key ingredient is to construct a self-similar blowup solution to a related aggregation equation as an approximate solution, with subcritical scaling relative to the original model. Based on this construction, we employ a robust weighted $L^2$ method to prove the stability of this approximate solution, where modulation ODEs are introduced to enforce local vanishing conditions for the perturbation lying in a singular-weighted $L^2$ space. As a byproduct, we exhibit a new family of type I blowup mechanisms for the classical 3D Keller-Segel equation.

Figures

Figures reproduced from arXiv: 2504.12231 by the authors.

Figure 1
Figure 1. QfQ-plane with P0 above the line fQ = β. In this case, there is only a trivial solution solving (3.3) with (Q(0), fQ(0)) = P0. We remark here that in Case I β < 1 3(1−µ)  , motivated by the related phase portrait (see [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. QfQ-plane with P0 on the line fQ = β. In this case, though it seems possible to have a solution curve connecting between P0 and the origin O from the phase portrait, with the overwhelming singularity on the right￾hand side of (3.3), after some careful analysis, the only possible smooth solution starting from P0 to should be the trivial one: (Q(r), fQ(r)) ≡ P0 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. QfQ-plane with P0 below fQ = β. In this case, we can always choose countably many {βj}j (see Theorem 1.1 and Lemma 3.2 for more details) such that there exists a smooth solution curve (the red curve in the figure) lying in M and connecting P0 and the origin O simultaneously, we will discuss this case in more details later in Lemma 3.2. 1.3.3. Stability analysis beyond the spectral theory. After selecting the self-si… view at source ↗

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  1. The traveling wave solutions of the 1D hyperbolic Keller-Segel equations

    math.AP 2026-08 conditional novelty 6.0 of 10

    Explicit families of entropy-admissible stationary and traveling wave solutions are constructed and classified for the 1D hyperbolic Keller-Segel system with quorum sensitivity.

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