{
  "abstract": "In this paper, we establish a sharp non-uniqueness result to the semilinear wave equation \\[ \\partial_{tt}u-\\partial_{xx} u= Θ_1(\\partial_t u)^2 + Θ_2(\\partial_x u)^2. \\] For any $Θ_1 Θ_2 < 0$, $β<1$, we construct infinitely many weak solutions $u \\in C_{t,x}^β$ with zero initial data. To our knowledge, this is the first non-uniqueness result for a semilinear wave equation. At this low level of regularity $C_{t,x}^β(β<1)$, the quadratic term is defined by Littlewood--Paley decomposition in $C_t H_x^{-s}$. Our construction is based on a frequency-localized convex integration scheme. Together with the classical local uniqueness theory in $C_{t,x}^{1}$, this non-uniqueness result is sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15442",
  "author_total": 2,
  "authors": [
    {
      "name": "Peng Qu"
    },
    {
      "name": "Mingxin Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3gahxg6w06hfj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15442",
  "fulltext_url": null,
  "id": "3gahxg6w06hfj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15442",
  "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",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7213244561",
  "pdf_url": "https://arxiv.org/pdf/2609.15442",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3gahxg6w06hfj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15442",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15442"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp non-uniqueness of weak solutions to the semilinear wave equation",
  "title_text": "Sharp non-uniqueness of weak solutions to the semilinear wave equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/3gahxg6w06hfj",
  "venue": "arXiv",
  "versions": []
}