{
  "abstract": "Consider positive classical solutions of the Cauchy problem for $u_t-Δu=u^p-μ|\\nabla u|^q$, with sufficiently smooth initial data $u_0$, in the strong absorption regime, namely $q\\ge p>1$ and $μ>0$. It has remained a long standing open problem whether finite time blow-up can occur if $u_0$ decays at space infinity. We here solve this problem, showing that the answer in negative. To this end, we establish a novel, universal Bernstein-type gradient estimate, of independent interest, valid only in the range $q\\ge p$. Moreover, the decay assumption on $u_0$ turns out to be essentially optimal. Indeed we show that the solution blows up in finite time whenever $u_0$ has a positive limit at infinity, and we determine the blow-up behavior of $u$ at space infinity under an additional assumption.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28913",
  "author_total": 2,
  "authors": [
    {
      "name": "Philippe Souplet"
    },
    {
      "name": "Berikbol T. Torebek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0tq17kvbjgn7r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28913",
  "fulltext_url": null,
  "id": "0tq17kvbjgn7r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28913",
  "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": "W7223243319",
  "pdf_url": "https://arxiv.org/pdf/2610.28913",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0tq17kvbjgn7r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.28913",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28913"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Strong gradient absorption and initial decay prevent blow-up",
  "title_text": "Strong gradient absorption and initial decay prevent blow-up",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/0tq17kvbjgn7r",
  "venue": "arXiv",
  "versions": []
}