{
  "abstract": "We consider the Cauchy problem $$u_t-Δu=|u|^p,\\qquad (t,x)\\in(0,T)\\times\\R^n, \\quad u(0,x)=\\eps u_0(x),\\qquad x\\in\\R^n,$$ with small sign-changing initial data having zero total mass. We assume $$u_0\\in L^1(\\R^n)\\cap L^\\infty(\\R^n),\\quad |x|u_0\\in L^1(\\R^n),\\quad \\int_{\\R^n}u_0(x)\\,dx=0,$$ and, for the sharp subcritical upper bounds, that $\\int_{\\R^n}x\\,u_0(x)\\,dx\\neq0.$ Let $T_\\eps$ denote the maximal lifespan. We identify a new transition exponent $p_m=1+\\frac1{n+1}$ inside the Fujita range $1<p\\le p_F:=1+2/n$, and establish the sharp lifespan estimates $$T_\\eps\\asymp \\begin{cases} \\eps^{-\\frac{2(p-1)}{2-(n+1)(p-1)}}, &1<p<p_m,\\\\[2mm] \\eps^{-\\frac2{n+1}} \\bigl(\\log\\frac1\\eps\\bigr)^{-\\frac2{n+2}}, &p=p_m,\\\\[2mm] \\eps^{-p\\left(\\frac{1}{p-1}-\\frac{n}{2}\\right)^{-1}}, &p_m<p<p_F,\\\\[2mm] \\exp\\!\\left(\\eps^{-p(p-1)}\\right), & p=p_F. \\end{cases}$$ These estimates reveal a lifespan phenomenon that is different from the classical Lee--Ni law for initial data with positive mass. In the zero-mass setting, the leading linear contribution is dipole-like, while the nonlinear source subsequently generates a positive mass of size \\(O(\\eps^p)\\). Their competition produces the additional threshold \\(p_m\\), the logarithmic correction at $p=p_m$, and, at $p=p_F$, a substantially longer critical lifespan with exponent $p_F(p_F-1)$ instead of the classical Lee--Ni exponent $p_F-1$. The upper estimates are obtained by backward-Gaussian and critical scale-ODE test-function arguments, whereas the lower estimates follow from a unified $L^1$--$L^\\infty$ bootstrap preserving the zero-mass cancellation of the linear flow and controlling the mass generated by the nonlinear source. Thus, although $p_F$ remains the Fujita critical exponent, zero initial mass creates a new quantitative lifespan regime below and at the critical exponent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17318",
  "author_total": 1,
  "authors": [
    {
      "name": "Berikbol T. Torebek"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8tg8396xgq3vj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17318",
  "fulltext_url": null,
  "id": "8tg8396xgq3vj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17318",
  "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/2609.17318",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8tg8396xgq3vj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17318",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17318"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "New sharp lifespan estimates for a semilinear heat equation with zero-mass sign-changing initial data",
  "title_text": "New sharp lifespan estimates for a semilinear heat equation with zero-mass sign-changing initial data",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/8tg8396xgq3vj",
  "venue": "arXiv",
  "versions": []
}