{
  "abstract": "We study the space-time concentration or blow-up asymptotics of radially decreasing solutions of the parabolic-elliptic Keller-Segel system in the whole space or in a ball. We show that, for any { blow-up} solution in dimensions $3\\le n\\le 9$ (assuming finite mass in the whole space case), there exists a nonflat backward self-similar solution $U$ such that $$u(x,t)=(1+o(1))U(x,t),\\quad\\hbox{as $(x,t)\\to (0,T)$,}$$ { where $T$ is the blow-up time.} This macroscopic behavior is important from the physical point of view, since it gives a sharp description of the concentration phenomenon in the scale of the original space-time variables~$(x,t)$. It strongly improves on existing results, since such behavior was previously known ([Giga-Mizoguchi-Senba]) to hold only in the microscopic scale $|x|\\le O(\\sqrt{T-t})$ as $t\\to T$ (and in the whole space case only). As a consequence, we obtain the two-sided global estimate $$C_1\\le (T-t+|x|^2)u(x,t)\\le C_2\\quad\\hbox{in $B_R\\times(T/2,T)$},$$ whose upper part only was known before ([Souplet-Winkler]), as well as the sharp final profile: $$\\lim_{x\\to 0} |x|^2u(x,T)=L\\in(0,\\infty).$$ The latter improves, with a different proof, the recent result of [Bai-Zhou] by excluding the possibility $L=0$. We also give extensions of these results, in higher dimensions, to type~I and to time monotone solutions. Moreover, we extend the known results on type I estimates and on convergence in similarity variables, and significantly simplify their proofs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2601.14469",
  "author_total": 2,
  "authors": [
    {
      "name": "Loth Damagui Chabi"
    },
    {
      "name": "Philippe Souplet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0pq91qn3wa2cx.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "0pq91qn3wa2cx",
  "issns": [],
  "landing_url": "https://hal.science/hal-05801885v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05801885/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0pq91qn3wa2cx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:58Z",
      "native_id": "hal-05801885",
      "source": "hal",
      "url": "https://hal.science/hal-05801885v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3",
  "title_text": "Sharp macroscopic blow-up behavior for the parabolic-elliptic Keller-Segel system in dimensions n ≥ 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:58Z",
  "url": "https://research.draconitic.com/paper/0pq91qn3wa2cx",
  "venue": "HAL",
  "versions": []
}