{
  "abstract": "In this article, we consider parabolic equations of the form $$\\partial_t u(x,t)=\\Delta u(x,t) - Bu(x,t) + F(u(x,t))$$ where $u$ is valued in a transverse Hilbert space $Y$ and $B$ is a positive self-adjoint operator on $Y$, which allows for a different diffusion mechanism in the transverse direction. Our aim is to consider solutions with infinite energy and we will study the Cauchy problem in the uniformly local spaces associated with the norm $$\\|u\\|_{L^2_\\ul(\\RR,Y)}= \\sup_{a\\in\\RR} \\|u\\|_{L^2(B(a,1),Y)}.$$ For the classical parabolic equation, i.e. when $Y=\\RR$, it is known that the Cauchy problem is ill-posed in the weak version of the uniformly local spaces but well-posed in a stronger version, where additional uniform continuity is required. In this paper, we show that the linear operator $\\partial^2_{xx} - B$ is not necessarily a sectorial operator in any version of the uniformly local Lebesgue space, due to the possible non-density of its domain. We then use the theory of parabolic abstract evolution equations to formulate a well-posed Cauchy problem, even in the weak version of the uniformly local space. In particular, we believe that this paper offers a new perspective on the comparison between the two versions of the uniformly local spaces and also provides a new natural example of differential operators with non-dense domain.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Romain Joly"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1v3t2pkqxf084.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "1v3t2pkqxf084",
  "issns": [
    "1259-1734",
    "2118-7436"
  ],
  "landing_url": "https://hal.science/hal-05830103v1",
  "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"
    ],
    [
      "hal",
      "1.math.math-ap"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05830103/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": "Université Blaise-Pascal - Clermont-Ferrand",
  "ris": "https://research.draconitic.com/paper/1v3t2pkqxf084.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:00Z",
      "native_id": "hal-05830103",
      "source": "hal",
      "url": "https://hal.science/hal-05830103v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-ap",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces",
  "title_text": "Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:18:00Z",
  "url": "https://research.draconitic.com/paper/1v3t2pkqxf084",
  "venue": "Annales Mathématiques Blaise Pascal",
  "versions": []
}