{
  "abstract": "This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(Ω)$ and Hölder stability from terminal-time observations for sources in a suitable spectral Gevrey class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10071",
  "author_total": 3,
  "authors": [
    {
      "name": "Ze Li"
    },
    {
      "name": "Zhiyuan Li"
    },
    {
      "name": "Zhi Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/688h470pg4zep.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10071",
  "fulltext_url": null,
  "id": "688h470pg4zep",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10071",
  "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.SP"
    ],
    [
      "openalex",
      "2610|Numerical methods in inverse problems"
    ]
  ],
  "openalex": "W7212142439",
  "pdf_url": "https://arxiv.org/pdf/2609.10071",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/688h470pg4zep.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.10071",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10071"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension",
  "title_text": "Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/688h470pg4zep",
  "venue": "arXiv",
  "versions": []
}