{
  "abstract": "Let $f(t,x),M(t,x)\\in C([0,1]^2)$ with $M(t,x)>0$. We consider differential equations of the form \\[ \\frac{\\partial f}{\\partial t}(t,x)=\\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},\\quad x>0. \\] For a fixed positive weight $M$, we ask whether the condition $f(0,x)=0$ forces $f\\equiv 0$. We show the answer is negative for smooth functions: there exist $f(t,x),M(t,x)\\in C^{\\infty}([0,1]^2)$ with $f(0,x)=0$, $f(t,0)\\not\\equiv 0$, and $M(t,x)>0$ satisfying the above equation. However, we show that for a Borel-Laplace class of $M(t,x)$, the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.04810",
  "author_total": 1,
  "authors": [
    {
      "name": "Brian Street"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0s9b6a9se1e0m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.04810",
  "fulltext_url": null,
  "id": "0s9b6a9se1e0m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.04810",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Differential Equations and Boundary Problems"
    ]
  ],
  "openalex": "W7160511961",
  "pdf_url": "https://arxiv.org/pdf/2605.04810",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0s9b6a9se1e0m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2605.04810",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.04810"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Non-uniqueness for a differential equation",
  "title_text": "Non-uniqueness for a differential equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/0s9b6a9se1e0m",
  "venue": "arXiv",
  "versions": []
}