{
  "abstract": "This paper studies the spectral properties and nonlinear asymptotic stability of a family of finite-time self-similar blow-up solutions to the three-dimensional Keller--Segel system constructed by matching interior and exterior profiles within the framework of matched asymptotic expansions. For every sufficiently large matching index $n$, the full linearized operator around the stationary state $U_n$ in self-similar variables is analyzed on $L^2(\\mathbb R^3)$. Sturm zero counting in the radial mode, a wave operator that reduces the nonlocal $l=1$ equation to a local equation, and a Mellin--Newton quadratic form for all $l\\ge2$ show that, after the scaling and translation modes and the finitely many genuinely unstable radial modes are removed, the remaining spectrum is separated from the imaginary axis by a positive distance. In addition, the optimal lower bound on the real parts of the spectrum is $1/4$ in every mode $l\\ge2$. On the stable subspace, an exponentially decaying semigroup and a modified energy equivalent to the $L^2$ norm are constructed, and the logarithmic asymptotic decay rate of the semigroup norm is proved to equal the stable spectral gap. Finally, modulation equations, $H^2$ energy estimates, control of the scaling derivative, and Brouwer's no-retraction theorem yield nonradial nonlinear asymptotic stability of the corresponding self-similar blow-up solutions after the initial coefficients in the finitely many unstable radial directions have been chosen suitably.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06479",
  "author_total": 3,
  "authors": [
    {
      "name": "Te Li"
    },
    {
      "name": "Yuwei Sun"
    },
    {
      "name": "Kaiqiang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8r2c6y0f8jsqp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06479",
  "fulltext_url": null,
  "id": "8r2c6y0f8jsqp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06479",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7212050682",
  "pdf_url": "https://arxiv.org/pdf/2609.06479",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8r2c6y0f8jsqp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.06479",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06479"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions",
  "title_text": "Optimal Spectral Lower Bounds and Nonradial Nonlinear Asymptotic Stability of a Family of Three-Dimensional Keller--Segel Self-Similar Blow-Up Solutions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/8r2c6y0f8jsqp",
  "venue": "arXiv",
  "versions": []
}