{
  "abstract": "At the Fujita exponent $p=1+2/N$ we construct a strictly positive mild solution of $u_t-Δu=|u|^{p-1}u$ in $\\mathbb R^N$ whose initial trace in $L^1$ is zero. In similarity variables the solution approaches a Gaussian profile, and $$\\|U(t)\\|_1\\sim C_N(\\log(1/t))^{-N/2},\\qquad \\|U(t)\\|_\\infty\\sim c_Nt^{-N/2}(\\log(1/t))^{-N/2},$$ with explicit constants $C_N$ and $c_N$. In particular, uniqueness fails at the borderline logarithmic rate, and the solution lies outside every stronger logarithmic uniqueness class; this answers two questions raised by Miyamoto [J. Evol. Equ. 21 (2021), 151--166, Remark~1.6]. We also prove that, up to time translation, this is the only ancient solution of fixed sign of the rescaled equation that decays in a Gaussian weighted supremum norm. On a bounded smooth domain the whole-space solution can be localized at finitely many interior points, with independently chosen signs, on top of an arbitrary bounded initial datum. The resulting solutions are classical for $t>0$ and continuous at $t=0$ in $L^1$. Consequently, uniqueness fails for every bounded datum, and positivity is lost for every bounded nonnegative datum, $u_0\\equiv1$ included. These results provide affirmative answers to Open Problems~3 and~4 posed by Brezis and Cazenave in [J. Anal. Math. 68 (1996), 277--304].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.40315",
  "author_total": 1,
  "authors": [
    {
      "name": "Berikbol T. Torebek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/amrh6fq7h00qa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.40315",
  "fulltext_url": null,
  "id": "amrh6fq7h00qa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.40315",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.40315",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/amrh6fq7h00qa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.40315",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.40315"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Nonuniqueness and failure of positivity preservation for doubly critical semilinear heat equations",
  "title_text": "Nonuniqueness and failure of positivity preservation for doubly critical semilinear heat equations",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/amrh6fq7h00qa",
  "venue": "arXiv",
  "versions": []
}