{
  "abstract": "We study finite-time blowup for the parabolic--elliptic Keller--Segel system on $\\mathbb{R}^d$ in dimensions $d\\geq11$, where the problem is mass supercritical. For every integer $l\\geq2$, we construct smooth radially symmetric solutions whose radial mass variable concentrates the normalized stationary state $Q$ at a quantized scale. Each blowup regime can be realized by solutions with nonnegative population density throughout their classical lifespan. More precisely, near the blowup time $T$, \\[ u(t,r)=\\frac{1}{λ^2(t)}\\left[Q\\left(\\frac{r}{λ(t)}\\right) +ε\\left(t,\\frac{r}{λ(t)}\\right)\\right], \\qquad λ(t)=c(T-t)^{\\frac{l}{γ(d)}}(1+o(1)), \\] where $c>0$ and $γ(d)=\\frac12\\bigl(d-2-\\sqrt{(d-2)(d-10)}\\bigr)$. The remainder $ε$ converges to zero in local $L^\\infty$ norms and in a range of high-order homogeneous Sobolev norms. Since $2l>γ(d)$, the concentration scale is strictly smaller than the parabolic scale $\\sqrt{T-t}$, and the resulting blowup is of type II. The $l$-th regime has exactly $l-1$ unstable radial modulation directions and is stable within a codimension-$(l-1)$ class of suitably regular radial initial data. The proof combines a generalized-kernel expansion driven by the algebraic tail of $Q$, modulation analysis, coercive weighted high-order energy estimates, and a finite-dimensional topological argument. This yields a quantized hierarchy of stationary-state concentration rates for the high-dimensional Keller--Segel flow.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16467",
  "author_total": 4,
  "authors": [
    {
      "name": "Thomas Y. Hou"
    },
    {
      "name": "Xiang Qin"
    },
    {
      "name": "Peicong Song"
    },
    {
      "name": "Zirui Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fnexp0957cska.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16467",
  "fulltext_url": null,
  "id": "fnexp0957cska",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16467",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213409991",
  "pdf_url": "https://arxiv.org/pdf/2609.16467",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fnexp0957cska.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16467",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16467"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Stability of Type II Blowup for the Keller-Segel System in High Dimensions",
  "title_text": "On the Stability of Type II Blowup for the Keller-Segel System in High Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/fnexp0957cska",
  "venue": "arXiv",
  "versions": []
}