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Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system

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Pith's one-line read This paper constructs finite-time type II blow-up solutions of the three-dimensional axially symmetric Keller-Segel system in which mass concentrates along any prescribed finite collection of rings, with the refined two-dimensional blow-up

desk verdict Substantial construction with a genuinely new potential expansion, but the advertised multi-ring theorem is only sketched and the outer barrier has a load-bearing gap; worth refereeing. read the letter →

arxiv 2508.08103 v1 pith:YAIYG3K3 submitted 2025-08-11 math.AP

Federico Buseghin , Juan Dávila , Manuel del Pino , Monica Musso This is my paper · ORCID
classification math.AP MSC 35B4435K5535Q92
keywords Keller-Segelsystemfinite-timeblow-uptypeIIaxiallysymmetricringconcentrationgluingmethodellipticintegralschemotaxis
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that the three-dimensional axially symmetric Keller-Segel system admits finite-time type II blow-up solutions for any prescribed finite collection of rings. The main theorem states that for $k$ distinct points $(r_1,z_1),\ldots,(r_k,z_k)\in\mathbb{R}_+\times\mathbb{R}$ there is an initial datum whose solution is, up to small errors, a sum of $k$ rescaled copies of the two-dimensional stationary profile $U(y)=8/(1+|y|^2)^2$, each centered at $(r_j(t),z_j(t))$ converging to the prescribed point, with the refined law $\lambda_j(t)\sim 2e^{-(\gamma+2)/2}\sqrt{T-t}\,e^{-\sqrt{|\ln(T-t)|/2}}$. This generalizes the single-ring construction of [52] and matches the refined rates known in two dimensions. The proof uses an inner-outer gluing scheme whose key novelty is a precise asymptotic expansion of the three-dimensional axially symmetric Newtonian potential of $U$, expressed through complete elliptic integrals, together with a four-term outer barrier that absorbs the transition from the two-dimensional near field to the genuinely three-dimensional far field.

What carries the argument

The carrying mechanism is an inner-outer gluing scheme around the ansatz $u_1=(\alpha/\lambda^2) U((r,z)-\xi)/\lambda)\,\chi+\phi_\lambda$. The inner problem is a parabolic perturbation of the two-dimensional linearized operator $L[\phi]=\Delta\phi-\operatorname{div}(U\nabla\psi)-\operatorname{div}(\phi\nabla\Gamma_0)$, solved after the time rescaling $\tau=\int_0^t ds/\lambda^2(s)$; the outer problem is solved by an explicit four-term barrier. The load-bearing three-dimensional input is the expansion of $\nabla v_0$: near the ring it is $\nabla\Gamma_0$, in an intermediate shell it is $\omega_1\nabla\Gamma_0+O(|\ln|(r,z)-\xi||)$, and in the far field it is $-4\pi q_1 (r,z)/|(r,z)|^3$. The e

What would settle it

Evaluate the quantities in (5.4)-(5.9) at points with $|(r,z)-\xi|=(T-t)^{1/4}$ using the explicit expansion (2.28); if the $O(|\ln(|(r,z)-\xi|)|)$ term from the intermediate regime is not dominated uniformly by $e^{-a\sqrt{2|\ln((T-t)+|(r,z)-\xi|^2)|}}\big((T-t)+|(r,z)-\xi|^2\big)^{-2}$ for any admissible $a$, the barrier argument of Theorem 5.1 fails.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for any $k$ distinct points $(r_1,z_1),\ldots,(r_k,z_k)\in\mathbb{R}_+\times\mathbb{R}$ there is an initial datum whose solution is $u=\sum_j \lambda_j^{-2} U((r,z)-(r_j,z_j))/\lambda_j)(1+o(1))$, uniformly on bounded sets, with $\lambda_j\sim 2e^{-(\gamma+2)/2}\sqrt{T-t}\,e^{-\sqrt{|\ln(T-t)|/2}}$ and centers converging to the prescribed points. The blow-up is type II and each ring carries the two-dimensional stationary profile $U$. The novelty is the multi-ring generalization with the refined two-dimensional rate, enabled by a precise expansion of the 3D Newtonian potential of $U$ and a barrier controlling the 2D-to-3D crossover.

Load-bearing premise

The load-bearing premise is that the four-term outer barrier can absorb, for sufficiently small $T$, the $O(|\ln(|(r,z)-\xi|)|)$ error from the intermediate regime of Proposition 2.10 where the gradient of $v_0$ passes from the two-dimensional to the three-dimensional form; if that estimate fails, the outer fixed point and the whole construction collapse.

Editorial extensions

If this is right

  • For every $k$ and every choice of $k$ distinct ring centers in $\mathbb{R}_+\times\mathbb{R}$, there is an initial datum producing finite-time blow-up with one concentration ring at each prescribed center.
  • The blow-up rate at each ring is the refined two-dimensional one, $\lambda_j(t)=2e^{-(\gamma+2)/2}\sqrt{T-t}\,e^{-\sqrt{|\ln(T-t)|/2}}(1+o(1))$, so the singularity is of type II.
  • Near each ring the asymptotic profile is the two-dimensional stationary state $U$, not a three-dimensional self-similar profile.
  • The inverse-Laplacian expansion in cylindrical coordinates provides a quantitative bridge between the two-dimensional and three-dimensional regimes and can be reused in other axisymmetric problems, such as vortex ring dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural test of the gluing's robustness is to track the first interaction between two rings at fixed separation as $T\to0$: the paper treats mixed terms as negligible, so a sharper estimate of their contribution would reveal a spacing condition under which the construction still closes.
  • If the same barrier strategy transfers, analogous multi-bubble type II solutions should exist for other supercritical drift-diffusion systems whose linearized operator around the stationary profile has the same neutral modes.
  • The paper establishes existence but not stability; a natural follow-up, not addressed here, is whether each multi-ring configuration is stable or unstable under axially symmetric perturbations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper constructs axially symmetric finite-time Type II blow-up solutions to the three-dimensional Keller–Segel system, concentrating mass along an arbitrary number of rings. The main result, Theorem 1.1, asserts that for any k distinct points in R_+ × R there is an initial datum whose solution has the multi-ring expansion u(r,z,t) = Σ_j λ_j(t)^{-2} U(((r,z)-(r_j(t),z_j(t)))/λ_j(t))(1+o(1)) with λ_j(t) ≈ 2e^{-(γ+2)/2} √(T-t) e^{-√(|ln(T-t)|/2)}. The proof combines a precise asymptotic expansion for the three-dimensional axially symmetric inverse Laplacian of U (Section 2, Theorems 2.9 and Proposition 2.10), a tailored correction φ_λ (Section 3), an inner–outer gluing system (Section 4), an outer theory with a four-term barrier (Theorem 5.1), and a fixed-point argument (Section 6). The multi-ring case is only sketched in Section 6.1, and several key propositions are imported from the authors' previous works [10] and [29].

Significance. If the construction is complete, the result is significant: it gives the first rigorous multi-ring Type II blow-up for the three-dimensional axisymmetric Keller–Segel system, with a refined blow-up rate consistent with the two-dimensional theory, thereby generalizing the single-ring result of Hou–Nguyen–Song. The single-ring expansion in Section 2 is carefully developed and contains original elliptic-integral analysis; the derivation of Theorem 2.9 and Proposition 2.10 is detailed and appears sound. The paper also provides explicit estimates for the gradient of the Newtonian potential in cylindrical coordinates, which are of independent use. However, the proof of the main theorem as written is incomplete at several load-bearing points: the outer barrier verification is partly verbal, the parameter choice in Proposition 4.2 is deferred to [10], and the multi-ring statement is only sketched. These gaps currently prevent the paper from meeting the standard of a fully verified construction.

major comments (4)
  1. [Section 5, Theorem 5.1 and (5.3)–(5.9)] The outer theory is load-bearing for the fixed-point argument in Section 6, but its proof is incomplete. In the intermediate region 2√δ(T−t) ≤ |(r,z)−ξ| ≤ ε, Proposition 2.10 gives ∇v0 = ω1 ∇Γ0 + O(|ln|(r,z)−ξ||). The proof of Theorem 5.1 replaces ∇v0 by the piecewise expression (5.3) and states that the discarded terms 'can be absorbed into parts of the barrier'. The estimate below (5.3) bounds the drift-error contribution by √T |ln T| times the main singularity, but it does not display a uniform pointwise domination of the full remainder, including the O(|ln|) term and the (ω1−1)∇Γ0 contribution, in the overlap with the region where φ3 is active. The proof ends with 'rescaling and applying standard elliptic estimates'. A rigorous, explicit absorption estimate in this intermediate zone is needed; without it, the linear outer operator T_o^p and hence the contraction argument in Section 6
  2. [Section 6.1] Theorem 1.1 is stated for arbitrary k distinct points, but the multi-ring proof is only sketched. The text says 'We omit most of the details' and discusses only the case k=2. For k=2, the interaction terms E_{1,2} and E_{2,1} are asserted to be negligible without estimates, and the resulting two inner equations, two outer equations, orthogonality conditions, and parameter corrections are not written down. No induction or general construction for k distinct rings is provided. Since the multi-ring statement is the paper's main advertised generalization, this is a central gap that must be addressed by giving the full construction or a precise reduction showing that the k-ring case follows from the single-ring one with explicit control of inter-ring interactions.
  3. [Section 4.1, Proposition 4.2] Proposition 4.2 is the mechanism that selects λ0(t) and α0(t) and produces the refined blow-up rate quoted in Theorem 1.1. Its proof is almost entirely deferred: the text states that after some reductions 'the detailed proof follows directly from the arguments presented in Section 4 of [10]'. The orthogonality conditions (4.15)–(4.16) are central to the construction, and the asymptotic expression for λ0(t) is a claimed improvement over [52]. At minimum, the nonlocal equation for λ0˙λ0(t), the role of the initial time −ε(T), and the solvability argument should be stated explicitly in the present setting. As written, the reader cannot verify the validity of (4.15)–(4.16) or the refined rate without consulting an external preprint.
  4. [Sections 3 and 4.5] The inner theory is imported from [10] as black boxes in Propositions 4.3–4.5, and Proposition 3.2 is proved only with the sentence 'For brevity, we omit the detailed computations.' These estimates are used in Section 6 to control the nonlocal terms involving ∇ψλ, so the adaptation to the axisymmetric three-dimensional setting must be demonstrated or precisely referenced. If the authors intend [10] to be used as a black box, they should state which specific statements and which modifications are needed; otherwise the present manuscript is not self-contained at a load-bearing point.
minor comments (5)
  1. [Section 5, Proposition 5.2] The hypothesis reads 'Assume that (2.6) and (2.6) are satisfied.' Presumably the second reference should be (2.7).
  2. [Throughout] There are several typographical errors: 'ecounter' near equation (2.17), 'iste ad' near the end of Section 5, and the reference [52] lists 'T. Y. How' instead of 'T. Y. Hou'.
  3. [Sections 5 and 6] The norms ∥g∥⋆,o, ∥ϕ∥⋆⋆,o, and ∥g∥o are used with slightly inconsistent notation; unify the definitions so the statements of Theorem 5.1 and the fixed-point setup in Section 6 are directly comparable.
  4. [Equation (4.37)] The expression uses φλ(T) although φλ is defined for t < T; clarify the intended meaning, e.g., a limit or an evaluation at a cut-off time.
  5. [Proposition 2.10] The O(|ln(|(r,z)−ξ|)|) term is singular as ξ is approached; it would be helpful to state explicitly that this means a bound by C|ln(|(r,z)−ξ|)| for small nonzero distance and to specify the constant's dependence on T, so that the absorption in Section 5 can be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the blow-up rate is fixed by solvability conditions, and the heavy self-citations are to independent technical tools, not to the target theorem.

full rationale

The paper is a construction paper rather than a data-fitting one. The blow-up rate is not fitted to the desired conclusion: λ0(t) is selected by the orthogonality conditions (4.15)–(4.16), which are solvability conditions for the linearized inner operator L, and Proposition 4.2 derives its asymptotic from a nonlocal equation rather than by assuming Theorem 1.1. The frequent citations to [10], [28], [29] are to prior technical machinery (two-dimensional inner theories, barrier constructions, and vortex gluing methods) that address different settings and do not assume the three-dimensional multi-ring theorem; they therefore function as independent support rather than as a self-referential chain. Theorem 2.9 and Proposition 2.10 derive the three-dimensional inverse-Laplacian expansion from elliptic-integral asymptotics with explicit error terms, and the outer barrier in Theorem 5.1 is assembled from four explicit functions with stated estimates. The proof's compressed or verbal absorption statements (e.g., 'can be absorbed', 'rescaling and applying standard elliptic estimates') are omissions of detail that would need a rigorous check, but they do not make any equation equal to its own input by construction, and no fitted parameter is renamed as a prediction. Thus the derivation chain is self-contained for the purposes of circularity analysis.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on importing the inner-outer gluing machinery from the authors' prior work [10] and [28], plus a new 3D Newtonian expansion. No new physical entities are introduced. The main axiom ledger entries are the quoted results from previous papers and the hand-chosen constants of the construction.

free parameters (4)
  • delta
    Small positive cutoff width in chi_0; fixed in Section 4.5 to control the B operator cutoff.
  • epsilon(T)
    Small initial time in (3.1); its role is described in Section 4.1 and it is needed for the orthogonality conditions.
  • tau_0
    Large initial rescaled time in (4.38) for the infinite-time inner problem.
  • mu, sigma, nu, rho, q, vartheta, a, b, m1, m2, m3
    Small and large constants fixing norms and cutoffs in Sections 4.3, 5, and 6; chosen by hand in the fixed point argument.
assumptions (6)
  • ad hoc to paper Inner solution maps from [10] (stated here as Propositions 4.3, 4.4, 4.5)
    These are quoted results with proofs in [10]; the present construction depends on them.
  • ad hoc to paper Solvability lemma L[phi] = h under mass and moment cancellations
    Lemma 4.1 is quoted from Lemma 5.1 of [28] and is the basis for choosing lambda_0 and alpha_0.
  • standard math Elliptic integral expansions (2.11) and (2.12)
    Taken from Byrd-Friedman [11]; used in the proof of Theorem 2.9.
  • standard math Newtonian potential asymptotics (Lemmas 2.1, 2.2 and Corollary 2.3)
    Elementary facts stated without proof.
  • domain assumption Parameter estimates (2.6) and (2.7)
    Assumed throughout the expansions and later established in Proposition 4.2.
  • ad hoc to paper Outer barrier absorbs the discarded errors in (5.3)
    Theorem 5.1 relies on absorbing errors E0, E1, E2, E3 into the barrier with the right constant choices; the verification is only sketched.

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Pith. "Pith review of Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system." pith.science (2026). https://pith.science/paper/YAIYG3K3

@misc{pith2026250808103,
  author       = {Pith},
  title        = {Pith review of: Finite-time blow-up for the three dimensional axially symmetric Keller-Segel system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YAIYG3K3}},
  note         = {Machine review of arXiv:2508.08103}
}
read the original abstract

We construct axially symmetric finite-time blow-up solutions to the three-dimensional Keller-Segel system. By adapting gluing techniques, we derive a precise asymptotic expansion for Type II singularities that generalizes the recent work of Hou, Nguyen, and Song. In our construction the mass concentrates along multiple rings and we obtain a refined expansion for the blow-up rate.

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Works this paper leans on

84 extracted references · 76 canonical work pages

  1. [10]

    Buseghin, J

    F. Buseghin, J. D´ avila, M. del Pino, M. Musso, Existence of finite time blow-up in Keller-Segel system , arXiv:2312.01475

    arXiv

  2. [29]

    D´ avila, M

    J. D´ avila, M. del Pino, M. Musso, and J. Wei,Gluing methods for vortex dynamics in Euler flows. Archive for Rational Mechanics and Analysis, 235, pp. 1467–1530(2020)

    work page 2020

  3. [52]

    T. Y. How, V. T. Nguyen, P. Song Axisymmetric type II blowup solutions to the three dimensional Keller-Segel system , arXiv:2502.19775

    work page Pith review arXiv

  4. [1]

    Abramovitz and I

    M. Abramovitz and I. A. Stegun Handbook of mathematical functions , New York: Dover. ISBN 978-0-486-61272-0

  5. [2]

    Biler Local and global solvability of some parabolic systems modelling chemotaxis , , Adv

    P. Biler Local and global solvability of some parabolic systems modelling chemotaxis , , Adv. Math. Sci. Appl., 8 (1998), pp. 715–743

    1998

  6. [3]

    Biler, G

    P. Biler, G. Karch, P. Lauren¸ cot, and T. Nadzieja The 8π-problem for radially symmetric solutions of a chemotaxis model in a disc . Topol. Methods Nonlinear Anal. 27 (2006), no. 1, 133–147

    2006

  7. [4]

    Biler, G

    P. Biler, G. Karch, P. Lauren¸ cot, and T. Nadzieja The 8π-problem for radially symmetric solutions of a chemotaxis model in the plane . Math. Methods Appl. Sci. 29 (2006), no. 13, 1563–1583

    2006

  8. [5]

    Blanchet On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher , S ´eminaire Laurent Schwartz—Equations aux d´eriv´ees partielles et applications

    A. Blanchet On the parabolic-elliptic Patlak-Keller-Segel system in dimension 2 and higher , S ´eminaire Laurent Schwartz—Equations aux d´eriv´ees partielles et applications. Ann ´ee 2011–2012, S ´emin. ´ Equ. D´eriv. Partielles, ´ Ecole Polytech., Palaiseau, 2013, pp. Exp. No. VIII, 26

    work page 2011

Show all 84 references
  1. [6]

    Blanchet, E

    A. Blanchet, E. A. Carlen, and J. A. Carrillo Functional inequalities, thick tails and asymptotics for the critical mass patlak–keller–segel model Journal of Functional Analysis, 262(5):2142–2230, 2012

    2012

  2. [7]

    Blanchet, J

    A. Blanchet, J. A. Carrillo, and N. Masmoudi Infinite time aggregation for a critical Patlak-Keller-Segel model in R2, Comm. Pure Appl. Math., 61(10):1449–1481, 2008

    2008

  3. [8]

    M. P. Brenner, P. Constantin, L. P. Kadanoff, A. Schenkel, and S. C. VenkataramaniDiffusion, attraction and collapse Nonlinearity 12 1071

  4. [9]

    Blanchet, J

    A. Blanchet, J. Dolbeault, and B. Perthame Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions , Electron. J. Differential Equations, pages No. 44, 32 pp. (electronic), 2006

    2006

  5. [11]

    P. F. Byrd, M. D. Friedman Handbook of Elliptic Integrals for Engineers and Physicists , Grundlehren der mathema- tischen Wissenschaften N.67 (Second Edition), 1971

    1971

  6. [12]

    Campos & J

    J. Campos & J. Dolbeault, A functional framework for the Keller-Segel system: logarithmic Hardy-LittlewoodSobolev and related spectral gap inequalities , C. R. Math. Acad. Sci. Paris, 350 (2012), pp. 949–954

    2012

  7. [13]

    Campos & J

    J. Campos & J. Dolbeault, Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane , Comm. Partial Differential Equations, 39 (2014), pp. 806–841

    2014

  8. [14]

    J. Campos Serrano Mod‘eles attractifs en astrophysique et biologie: points critiques et comportement en temps grand des solutions , PhD thesis, Th` ese de l’Universit` e Paris Dauphine, 2012

    2012

  9. [15]

    Carlen & M

    E. Carlen & M. Loss Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on Sn, Geom. Funct. Anal., 2 (1992), pp. 90–104

    1992

  10. [16]

    Carlen & A

    E. Carlen & A. Figalli, Stability for a GNS inequality and the log-HLS inequality, with application to the critical mass Keller-Segel equation, Duke Math. J., 162 (2013), pp. 579–625

    2013

  11. [17]

    Chavanis, Nonlinear mean field Fokker-Planck equations

    P .H. Chavanis, Nonlinear mean field Fokker-Planck equations. application to the chemotaxis of biological populations, , The European Physical Journal B, 62 (2008), pp. 179–208

    2008

  12. [18]

    P. H. Chavanis & C. Sire, Virial theorem and dynamical evolution of self-gravitating Brownian particles in an un- bounded domain. I. Overdamped models , Phys. Rev. E (3), 73 (2006), pp. 066103, 16

    2006

  13. [19]

    P. H. Chavanis & C. Sire, Virial theorem and dynamical evolution of self-gravitating Brownian particles in an un- bounded domain. II. Inertial models , Phys. Rev. E (3), 73 (2006), pp. 066104, 13

    2006

  14. [20]

    Childress & J

    S. Childress & J. K. Percus Nonlinear aspects of chemotaxis , Math. Biosci., 56 (3-4):217–237, 1981

    1981

  15. [21]

    Collot, T

    C. Collot, T. Ghoul, N. Masmoudi, and V.T. Nguyen Spectral analysis for singularity formation of the 2D Keller-Segel system, Annals of PDE, Vol.8, Issue 1, 2022

    2022

  16. [22]

    Collot, T

    C. Collot, T. Ghoul, N. Masmoudi, and V.T. Nguyen Refined description and stability for singular solutions of the 2d Keller-Segel system , Comm. Pure Appl. Math., Volume 75, Issue 7, pp. 1419-1516, 2021

    2021

  17. [23]

    Collot, T

    C. Collot, T. Ghoul, N. Masmoudi, and V.T. Nguyen Collapsing-ring blowup solutions for the Keller-Segel system in three dimensions and higher , arXiv:2112.15518, 2021

    2021 arXiv

  18. [24]

    Collot & K

    C. Collot & K. Zhang On the stability of type I self-similar blow-ups for the Keller-Segel system in three dimensions and higher ,arXiv:2406.11358

    arXiv

  19. [25]

    Corrias, B

    L. Corrias, B. Pertham, and H. Zaag Global solutions of some chemotaxis and angiogenesis systems in high space dimension Milan Journal of Mathematics, 72(1):1–28, Oct 2004

    2004

  20. [26]

    Corrias, V

    L. Corrias, V. Calvez, and M. A. Ebde Blow-up, concentration phenomenon and global existence for the keller–segel model in high dimension Communications in Partial Differential Equations, 37(4):561–584, 2012

    2012

  21. [27]

    Cort´ azar, M

    C. Cort´ azar, M. del Pino, and M. Musso Green ’s function and infinite-time bubbling in the critical nonlinear heat equation,Journal of the European Mathematical Society, Vol. 22, No. 1 pp. 283–344, 2019

    2019

  22. [28]

    D´ avila, M

    J. D´ avila, M. del Pino, Dolbeault J., M. Musso, and J. Wei, Infinite time blow-up in the patlak-keller-segel system: existence and stability. arXiv:1911.12417, 2023

    1911 arXiv

  23. [30]

    D´ avila, M

    J. D´ avila, M. del Pino, M. Musso, and J. Wei, Leapfrogging vortex rings for the three-dimensional incompressible Euler equations Communications on Pure and Applied MathematicsVolume 77, Issue 10 pp. 3843-3957

  24. [31]

    D´ avila, M

    J. D´ avila, M. del Pino, C. Pesce, and J. Wei, Blow-up for the 3-dimensional axially symmetric harmonic map flow into S2. Discrete and Continuous Dynamical Systems, Volume 39, Issue 12: 6913-6943(2019)

    2019

  25. [32]

    D´ avila, M

    J. D´ avila, M. del Pino, and J. Wei, Singularity formation for the two-dimensional harmonic map flow into S2, Inventiones Mathematicae, Volume 219, issue 2, pp. 345-466 (2019)

    2019

  26. [33]

    Dejak, P

    S. Dejak, P. Lushnikov, Y. Ovchinnikov, and I. Sigal, On spectra of linearized operators for Keller–Segel models of chemotaxis, , Physica D: Nonlinear Phenomena, 241 (2012), pp. 1245–1254

    2012

  27. [34]

    del Pino Bubbling blow-up in critical parabolic problems, in Nonlocal and nonlinear diffusions and interactions: new methods and directions , vol

    M. del Pino Bubbling blow-up in critical parabolic problems, in Nonlocal and nonlinear diffusions and interactions: new methods and directions , vol. 2186 of Lecture Notes in Math., Springer, Cham, 2017, pp. 73–116. FINITE-TIME BLOW-UP FOR THE THREE DIMENSIONAL AXIALLY SYMMETR...

    2017

  28. [35]

    del Pino, M

    M. del Pino, M. Musso, and J. Wei Infinite-time blow-up for the 3-dimensional energy-critical heat equation , Anal. PDE, 13 (2020), pp. 215–274

    2020

  29. [36]

    del Pino & J

    M. del Pino & J. Wei Collapsing steady states of the Keller-Segel system. , Nonlinearity 19 (2006), no. 3, 661–684

    2006

  30. [37]

    J. I. Diaz, T. Nagai, and J. -M. Rakatoson, Symmetrization techniques on unbounded domains: application to a chemotaxis system on RN , , J. Differential Equations, 145 (1998), pp. 156–183

    1998

  31. [38]

    Dolbeault & B

    J. Dolbeault & B. Perthame Optimal critical mass in the two-dimensional Keller-Segel model in R2, C. R. Math. Acad. Sci. Paris, 339(9):611–616, 2004

    2004

  32. [39]

    Dolbeault & C

    J. Dolbeault & C. Schmeiser, The two-dimensional Keller-Segel model after blow-up , Discrete and Continuous Dy- namical Systems, 25 (2009), pp. 109–121

    2009

  33. [40]

    Ega˜ na Fern´ andez & S

    G. Ega˜ na Fern´ andez & S. Mischler,Uniqueness and long time asymptotic for the Keller-Segel equation: the parabolic- elliptic case , Arch. Ration. Mech. Anal., 220 (2016), pp. 1159–1194

    2016

  34. [41]

    Ghoul & N

    T. Ghoul & N. Masmoudi, Minimal mass blowup solutions for the Patlak-Keller-Segel equation , Comm. Pure Appl. Math., 71(10):1957–2015, 2018

    1957

  35. [42]

    I. S. Gradshteyn & I. M. Ryzhik Table of Integrals, Series, and Products Mathematics of Computation vol. 20 iss. 96, 1966

    1966

  36. [43]

    M. A. Herrero & J. J. L. Vel´ azquez, A blow-up mechanism for a chemotaxis model , Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 24(4):633–683, 1997

    1997

  37. [44]

    M. A. Herrero & J. J. L. Vel´ azquez, Chemotactic collapse for the Keller-Segel model , J. Math. Biol. 35: 177 ¯D194 (1996)

    1996

  38. [45]

    M. A. Herrero & J. J. L. Vel´ azquez, Finite-time aggregation into a single point in a reaction-diffusion system , Non- linearity, 10(6): 1739–1754,1997

    1997

  39. [46]

    M. A. Herrero & J. J. L. Vel´ azquez, Singularity patterns in a chemotaxis model. , Math. Ann. 306 (1996), no. 3, 583–623

    1996

  40. [47]

    M. A. Herrero, E. Medina, and J. J. L. Vel´ azquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10(6): 1739–1754, 1997

    1997

  41. [48]

    M. A. Herrero, E. Medina, and J. J. L. Vel´ azquez, Self-similar blowup for a reaction-diffusion system , J. Comput. Appl. Math., 97(1-2):99– 119, 1998

    1998

  42. [49]

    Hillen & K

    T. Hillen & K. J. Painter, , A user’s guide to pde models for chemotaxis , Journal of Mathematical Biology, 58 (2008), pp. 183–217

    2008

  43. [50]

    Horstmann From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

    D. Horstmann From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I , Jahresber. Deutsch. Math.-Verein., 105 (2003), pp. 103–165

    1970

  44. [51]

    Horstmann From 1970 until present: the Keller-Segel model in chemotaxis and its consequences

    D. Horstmann From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. II , Jahresber. Deutsch. Math.-Verein., 106 (2004), pp. 51–69

    1970

  45. [53]

    J¨ ager & S

    W. J¨ ager & S. Luckhaus,On explosions of solutions to a system of partial differential equations modelling chemotaxis , Trans. Amer. Math Soc., 329 (1992), pp. 819-824

    1992

  46. [54]

    E. L. Kaplan Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36, 1948

    1948

  47. [55]

    Karp & S

    D. Karp & S. M. Sitnik Asymptotic approximations for the first incomplete elliptic integral near logarithmic singu- larity., J. Comput. Appl. Math. 205 (1), pp. 186–206., 2007

    2007

  48. [56]

    N. I. Kavallaris & P. Souplet Grow-up rate and refined asymptotics for a two-dimensional Patlak-KellerSegel model in a disk , , SIAM J. Math. Anal., 40 (2008/09), pp. 1852–1881

    2008

  49. [57]

    E. F. Keller Assessing the Keller-Segel model: how has it fared? , Biological growth and spread (Proc. Conf., Heidelberg, 1979), Lecture Notes in Biomath., vol. 38, pp. 379–387, Springer, Berlin-New York, 1980

    1979

  50. [58]

    E. F. Keller & L. A. Segel, Initiation of slime mold aggregation viewed as an instability , Journal of Theoretical Biology 26 (1970), no. 3, 399 – 415

    1970

  51. [59]

    E. F. Keller & L. A. Segel Model for chemotaxis , Journal of Theoretical Biology 30 (1971), no. 2, 225 – 234

    1971

  52. [60]

    E. F. Keller & L. A. Segel Traveling bands of chemotactic bacteria: A theoretical analysis , Journal of Theoretical Biology 30 (1971), no. 2, 235 – 248

    1971

  53. [61]

    Kozono & Y

    H. Kozono & Y. Sugiyama Local existence and finite time blow-up of solutions in the 2-D Keller-Segel system. J. Evol. Equ. 8 (2008), no. 2, 353–378

    2008

  54. [62]

    Z. Li & T. Zhou Nonradial stability of self-similar blowup to Keller-Segel equation in three dimensions arXiv:2501.07073

    arXiv

  55. [63]

    L´ opez-G´ omez, T

    J. L´ opez-G´ omez, T. Nagai, and T. YamadaThe basin of attraction of the steady-states for a chemotaxis model in R2 with critical mass , Arch. Ration. Mech. Anal., 207 (2013), pp. 159–184

    2013

  56. [64]

    L´ opez-G´ omez, T

    J. L´ opez-G´ omez, T. Nagai, and T. YamadaNon-trivial ω-limit sets and oscillating solutions in a chemotaxis model in R2 with critical mass , J. Funct. Anal., 266 (2014), pp. 3455–3507

    2014

  57. [65]

    Mizoguchi Refined asymptotic behavior of blowup solutions to a simplified chemotaxis system

    N. Mizoguchi Refined asymptotic behavior of blowup solutions to a simplified chemotaxis system. Communications on Pure and Applied Mathematics, n/a(n/a), 2020

    2020

  58. [66]

    Nagai Blow-up of radially symmetric solutions to a chemotaxis system

    T. Nagai Blow-up of radially symmetric solutions to a chemotaxis system . Adv. Math. Sci. Appl., 5(2):581–601, (1995)

    1995

  59. [67]

    Nagai Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains, Journal of Inequalities and Applications, 2001 (2001), p

    T. Nagai Blowup of nonradial solutions to parabolic–elliptic systems modeling chemotaxis in two-dimensional domains, Journal of Inequalities and Applications, 2001 (2001), p. 970292

    2001

  60. [68]

    Nagai Convergence to self-similar solutions for a parabolic-elliptic system of drift-diffusion type in R2, Adv

    T. Nagai Convergence to self-similar solutions for a parabolic-elliptic system of drift-diffusion type in R2, Adv. Differential Equations, 16 (2011), pp. 839–866

    2011

  61. [69]

    Nagai & T

    T. Nagai & T. Ogawa, Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in R2, Funkcial. Ekvac., 59 (2016), pp. 67–112

    2016

  62. [70]

    Nagai & T

    T. Nagai & T. Yamada Boundedness of solutions to a parabolic-elliptic Keller-Segel equation in R2 with critical mass, Adv. Nonlinear Stud., 18 (2018), pp. 337–360

    2018

  63. [71]

    Nanjundiah, Chemotaxis, signal relaying and aggregation morphology , Journal of Theoretical Biology, 42(1):63 – 105, 1973

    V. Nanjundiah, Chemotaxis, signal relaying and aggregation morphology , Journal of Theoretical Biology, 42(1):63 – 105, 1973. 36 F. BUSEGHIN, J. D ´AVILA, M. DEL PINO, AND M. MUSSO

    1973

  64. [72]

    V. T. Nguyen, N. Nouaili, and H. Zaag Construction of type I-log blow-up for the Keller-Segel system in dimension 3 and 4, arXiv:2309.13932

    arXiv

  65. [73]

    Nishihara, A note on the stability of travelling wave solutions of Burgers’ equation , Japan J

    K. Nishihara, A note on the stability of travelling wave solutions of Burgers’ equation , Japan J. Appl. Math., 2(1):27–35, 1985

    1985

  66. [74]

    C. S. Patlak, Random walk with persistence and external bias , Bull. Math. Biophys. 15 (1953), 311–338

    1953

  67. [75]

    Rapha¨ el & R

    P. Rapha¨ el & R. Schweyer,On the stability of critical chemotactic aggregation . Math. Ann., 359 (1-2): 268-377, 2014

    2014

  68. [76]

    Senba, Blowup behavior of radial solutions to J¨ ager-Luckhaus system in high dimensional domains , Funkcialaj Ekvacioj, 48(2):247–271, 2005

    T. Senba, Blowup behavior of radial solutions to J¨ ager-Luckhaus system in high dimensional domains , Funkcialaj Ekvacioj, 48(2):247–271, 2005

    2005

  69. [77]

    Senba & T

    T. Senba & T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis , J. Funct. Anal., 191 (2002), pp. 17–51

    2002

  70. [78]

    Sire & P

    C. Sire & P. -H. Chavanis, Thermodynamics and collapse of self-gravitating brownian particles in d dimensions , Phys. Rev. E, 66 (2002), p. 046-133

    2002

  71. [79]

    Souplet & M

    P. Souplet & M. Winkler, Blow-up profiles for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3, Comm. Math. Phys., 367 (2019), pp. 665–681

    2019

  72. [80]

    Van de Vel On the series expansion method for computing incomplete elliptic integrals of the first and second kinds., Math

    H. Van de Vel On the series expansion method for computing incomplete elliptic integrals of the first and second kinds., Math. Comp. 23 (105), pp. 61–69, 1969

    1969

  73. [81]

    J. J. L. Vel´ azquez,Stability of some mechanisms of chemotactic aggregation . SIAM J. Appl. Math., 62(5):1581–1633, 2002

    2002

  74. [82]

    J. J. L. Vel´ azquez,Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions . SIAM J. Appl. Math. 64 (2004), no. 4, 1198–1223

    2004

  75. [83]

    J. J. L. Vel´ azquez,Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions.. SSIAM J. Appl. Math. 64 (2004), no. 4, 1224–1248

    2004

  76. [84]

    Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations

    G. Wolansky, On steady distributions of self-attracting clusters under friction and fluctuations . Arch. Rational Mech. Anal.,119(4):355–391, 1992. F. Buseghin: Laboratoire AGM, CY Cergy Paris Universit ´e, 2 avenue Adolphe Chauvin, 95300 Pontoise, France Email address : feder...

    1992

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