{
  "abstract": "This paper establishes time decay estimates for the following two-dimensional beam (plate) equation with a decaying real-valued potential $V$: \\begin{equation*} \\partial_t^2 u + (Δ^2 + V) u = 0, \\qquad u(0,x)=f(x),\\quad \\partial_t u(0,x)=g(x). \\end{equation*} When zero is a regular point or a first-kind resonance of $H=Δ^2+V$, we first prove sharp $L^1\\to L^\\infty$ estimates for the solution operators: \\begin{align*} \\left\\|\\cos(t\\sqrt{H})P_{\\mathrm{ac}}(H)\\right\\|_{L^1\\to L^\\infty} + \\left\\|\\frac{\\sin(t\\sqrt{H})}{t\\sqrt{H}}P_{\\mathrm{ac}}(H)\\right\\|_{L^1\\to L^\\infty} \\lesssim \\frac{1}{|t|}, \\end{align*} and obtain an enhanced decay $(|t|\\log|t|)^{-1}$ in logarithmically weighted spaces $L^1_ω\\to L^\\infty_{-ω}$ with $ω(x)=\\log(2+|x|)$. For second-kind resonances of $H$ (the bi-Laplacian $Δ^2$ belongs to this class), a non-zero trace moment $\\langle |x|^2V,ϕ\\rangle\\neq0$ for some second-kind resonance function $ϕ$ induces severe threshold singularities, worsening the $L^1\\to L^\\infty$ estimate to $|t|^{-1}(\\log|t|)^2$. Finally, for third-kind resonances or a zero eigenvalue, we prove that the presence of $d$-wave resonance leads to the worst $L^1\\to L^\\infty$ decay rate $\\sim(\\log|t|)^{-1}$. Several improved estimates are also obtained without a $d$-wave resonance. In particular, in the pure eigenvalue case (i.e., neither $d$-wave nor $p$-wave resonance), both propagators recover the optimal unweighted $L^1\\to L^\\infty$ estimate $|t|^{-1}.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.16793",
  "author_total": 4,
  "authors": [
    {
      "name": "Shuangshuang Chen"
    },
    {
      "name": "Han Cheng"
    },
    {
      "name": "Zijun Wan"
    },
    {
      "name": "Xiaohua Yao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4wtcp8k4ntbpd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.16793",
  "fulltext_url": null,
  "id": "4wtcp8k4ntbpd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.16793",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7164887405",
  "pdf_url": "https://arxiv.org/pdf/2606.16793",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4wtcp8k4ntbpd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2606.16793",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.16793"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Decay estimates for the two-dimensional Beam equation with potentials",
  "title_text": "Decay estimates for the two-dimensional Beam equation with potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/4wtcp8k4ntbpd",
  "venue": "arXiv",
  "versions": []
}