{
  "abstract": "In this paper, we establish the well-posedness of Cauchy problems for weak solutions to second-order degenerate parabolic equations with a non-smooth, time-dependent degenerate elliptic part that includes both bounded and unbounded lower-order terms. The unbounded lower-order terms are allowed to lie in mixed time-space Lebesgue or even Lorentz spaces. Our notion of weak solutions is formulated under minimal assumptions. We prove the existence and uniqueness of a fundamental solution, which coincides with the associated evolution family for the homogeneous problem (i.e., with zero source term) and provides a representation formula for all weak solutions. We also establish $L^2$ off-diagonal estimates for the fundamental solution and derive Gaussian upper bounds under the weak assumption of Moser's $L^2$-$L^\\infty$ estimates for weak solutions. Our approach is purely variational and avoids any a priori regularity assumptions on weak solutions or regularization via smooth approximations. Two key ingredients are norm inequalities for fractional powers of the degenerate Laplacian, and a set of embeddings that ensure time continuity of weak solutions, extending the classical Lions regularity theorem and accommodating a wide class of source terms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2512.03978",
  "author_total": 1,
  "authors": [
    {
      "name": "Khalid Baadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3kvn0p4747zxx.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "3kvn0p4747zxx",
  "issns": [
    "1864-8266",
    "1864-8266"
  ],
  "landing_url": "https://hal.science/hal-05399830v2",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
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      "1.math.math-ap"
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  "pdf_url": "https://hal.science/hal-05399830/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": "Walter de Gruyter GmbH",
  "ris": "https://research.draconitic.com/paper/3kvn0p4747zxx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:50Z",
      "native_id": "hal-05399830",
      "source": "hal",
      "url": "https://hal.science/hal-05399830v2"
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  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ap",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms",
  "title_text": "On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:50Z",
  "url": "https://research.draconitic.com/paper/3kvn0p4747zxx",
  "venue": "Advances in Calculus of Variation",
  "versions": []
}