{
  "abstract": "This paper develops a general framework for the well-posedness, regularity, and strong approximation of the stochastic reaction--diffusion equation (SRDE) with superlinear drift and diffusion coefficients. We first extend the well-posedness results in \\emph{W. Liu and M. Röckner, J. Funct. Anal., 2902--2922, 2010} and \\emph{W. Liu, J. Differential Equations, 572--592, 2013} to the case of superlinear diffusion in the Gelfand triple \\(V \\hookrightarrow H \\hookrightarrow V^*\\), with \\(V\\) equipped with the norm \\(\\|\\cdot\\|_V\\), and derive a moment estimate by establishing a new Itô formula for \\(\\|X\\|_V^p\\) with general \\(p \\ge 2\\). We then apply this abstract result to the SRDE, establish higher spatial regularity \\(\\dot H^{1+γ}\\) for any \\(γ\\in [0,1]\\) whenever the initial datum lies in the same Sobolev space, and obtain temporal Hölder regularity. Finally, we construct a family of tamed finite element methods (tamed-FEMs) for the SRDE under general assumptions on the tamed functions, derive their long-time unconditional stability, and establish optimal strong convergence rates. To our knowledge, this is the first strong approximation result for SPDEs with superlinear diffusion coefficients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16979",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhihui Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f7prmbmyeq8kt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16979",
  "fulltext_url": null,
  "id": "f7prmbmyeq8kt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16979",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7213418181",
  "pdf_url": "https://arxiv.org/pdf/2609.16979",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f7prmbmyeq8kt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.16979",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16979"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation",
  "title_text": "Well-posedness, Regularity, and Strong Approximations of Superlinear Stochastic Reaction-Diffusion Equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/f7prmbmyeq8kt",
  "venue": "arXiv",
  "versions": []
}