{
  "abstract": "We study the critical behavior of the inhomogeneous semilinear heat equation \\[ u_t-Δu=|u|^p+w(x) \\qquad\\text{in }(0,\\infty)\\times\\mathbb{R}^N, \\] where \\(N\\geq3\\), \\(p>1\\), and \\(w\\in L^1(\\mathbb{R}^N)\\) is nontrivial and has zero total mass: \\[ \\int_{\\mathbb{R}^N}w(x)\\,dx=0. \\] This zero-mass case was posed as an open problem by Bandle, Levine, and Zhang in 2000 and has remained unresolved since then. We provide a complete answer to this problem by proving nonexistence of global weak solutions for \\[ 1<p\\leq\\frac{N}{N-2}. \\] Combined with the known supercritical existence result for sufficiently small data, this shows that the critical exponent separating the nonexistence and existence regimes remains \\[ p_c=\\frac{N}{N-2}, \\] the same as in the positive-mass case.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22458",
  "author_total": 2,
  "authors": [
    {
      "name": "Mohamed Jleli"
    },
    {
      "name": "Bessem Samet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ewp9dqm7df41y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22458",
  "fulltext_url": null,
  "id": "ewp9dqm7df41y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22458",
  "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": "W7214045897",
  "pdf_url": "https://arxiv.org/pdf/2609.22458",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ewp9dqm7df41y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22458",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22458"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Zero-Mass Problem of Bandle, Levine, and Zhang",
  "title_text": "On the Zero-Mass Problem of Bandle, Levine, and Zhang",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/ewp9dqm7df41y",
  "venue": "arXiv",
  "versions": []
}