{
  "abstract": "We address the question of global in time existence for the semilinear parabolic equation \\begin{equation*} \\left\\{ \\begin{array}{rcl} \\partial_t u-Δu&=&|u|^{b-1}u \\quad {\\rm ~in~} \\mathbb{R}^n\\times (0,\\infty),\\\\ u(\\cdot,0)&=& u_0 \\quad {\\rm ~in~} \\mathbb{R}^n, \\end{array} \\right. \\end{equation*} where $b> 1+2/n, n\\geq 1$. We found a space of initial data $Z$ that guarantees the existence of a global in time solution whenever the norm of the initial datum $u_0$ in $Z$ is small. The space $Z$ is optimal in the sense that if $u_0$ is a nonnegative distribution and the equation admits a nonnegative solution $u\\in L^b_{\\rm loc}(\\mathbb{R}^n\\times(0,\\infty))$, then $u_0\\in Z$. Moreover, our result contains all of the known results in the literature. In particular, it implies that global in time existence does hold for end-point Besov and Triebel-Lizorkin spaces, which resolves an open problem posed by L. Brandolese and F. Cortez (J. Funct. Anal. $\\textbf{276}$ (2019), 2589-2604).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27365",
  "author_total": 2,
  "authors": [
    {
      "name": "Gael Y. Diebou"
    },
    {
      "name": "Nguyen Cong Phuc"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9a7hr9emvsjtk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27365",
  "fulltext_url": null,
  "id": "9a7hr9emvsjtk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27365",
  "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",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7223087576",
  "pdf_url": "https://arxiv.org/pdf/2610.27365",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9a7hr9emvsjtk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27365",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27365"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Optimal global well-posedness for semilinear heat equations",
  "title_text": "Optimal global well-posedness for semilinear heat equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/9a7hr9emvsjtk",
  "venue": "arXiv",
  "versions": []
}