{
  "abstract": "Consider the heat equation: \\begin{equation}\\label{eq:00} \\begin{cases} \\displaystyle du(t,x)=\\frac12Δu(t,x)\\,dt+b(u(t,x))\\,dt, &t>0,\\ x\\in D,\\\\[1mm] u(t,x)=0,&t\\ge0,\\ x\\in\\partial D, u(0,x)=u_0(x),&x\\in D. \\end{cases} \\end{equation} It is well known that superlinear growth of the coefficient $b$ may lead to finite-time blow-up. In this paper, we show that, for every locally Lipschitz reaction coefficient $b:\\R\\to\\R$, there always exists a locally Lipschitz noise coefficient $σ:\\R\\to\\R$ such that the stochastic heat equation \\begin{equation}\\label{eq:01} \\begin{cases} \\displaystyle du(t,x)=\\frac12Δu(t,x)\\,dt+b(u(t,x))\\,dt+σ(u(t,x))\\,dB_t, &t>0,\\ x\\in D,\\\\[1mm] u(t,x)=0,&t\\ge0,\\ x\\in\\partial D, u(0,x)=u_0(x),&x\\in D. \\end{cases} \\end{equation} has a unique global solution for every $u_0\\in C_c(D)$. Here $(B_t)_{t\\ge0}$ is a real-valued Brownian motion. In particular, if $b=0$, the equation has a global solution for every locally Lipschitz $σ$, without any growth restriction at infinity. The proof uses reflected scalar barriers, a Lyapunov estimate, truncation, and comparison techniques.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13963",
  "author_total": 2,
  "authors": [
    {
      "name": "Yunfeng Chen"
    },
    {
      "name": "Tusheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7hd95w632q6xe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13963",
  "fulltext_url": null,
  "id": "7hd95w632q6xe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13963",
  "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"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7213346886",
  "pdf_url": "https://arxiv.org/pdf/2609.13963",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7hd95w632q6xe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2609.13963",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13963"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Non-blowup of stochastic heat equations by noise",
  "title_text": "Non-blowup of stochastic heat equations by noise",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/7hd95w632q6xe",
  "venue": "arXiv",
  "versions": []
}