{
  "abstract": "In this paper, we first establish a weighted bilinear fractional commutator estimate involving the spatial weight $\\langle x \\rangle^{2m}$. This estimate provides a powerful tool to resolve the difficulties arising from the nonlocality of the fractional Laplacian $(-Δ)^{σ/2}$ and the interaction between the unbounded weights and the nonlocal operator at infinity. We then apply this result to investigate the Cauchy problem for the fractional wave equation with time-dependent damping and source nonlinearity: $$ \\begin{cases} \\partial_t^2 u(t,x) + (-Δ)^{σ/2} u(t,x) + b(t) \\partial_t u(t,x) = |u(t,x)|^p, & (t,x) \\in (0,T) \\times \\mathbb{R}^N, \\\\ u(0,x) = \\varepsilon u_0(x), \\quad \\partial_t u(0,x) = \\varepsilon u_1(x), & x \\in \\mathbb{R}^N, \\end{cases} $$ where $σ\\in (0,2)$, the initial data scale with a small parameter $\\varepsilon > 0$, and the time-dependent damping coefficient $b(t)$ satisfies $b(t) \\approx (1+t)^{-β}$ for $β\\in [-1, 1)$. Utilizing scaling variables and constructing a weighted energy framework based on Theorem 1.1, we derive a priori differential inequalities for the total energy. For the supercritical case $p > p_c := 1 + \\fracσ{N}$, we prove the small-data global existence and uniqueness of strong solutions in weighted Sobolev spaces $H^{σ/2,m}(\\mathbb{R}^N) \\times H^{0,m}(\\mathbb{R}^N)$ by closing a bootstrap argument. For the subcritical ($1 < p < p_c$) and critical ($p = p_c$) cases, we establish precise lower bounds for the lifespan $T_\\varepsilon$ of blow-up solutions via delicate ODE analysis. Together with the corresponding upper estimates, our results fully determine the sharp lifespan estimates for all $β\\in [-1, 1)$, confirming that $p_c = 1 + \\fracσ{N}$ is indeed the critical exponent for the fractional wave equation with time-dependent damping.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.04247",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiayun Lin"
    },
    {
      "name": "Masahiro Ikeda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vv2mz32z449k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.04247",
  "fulltext_url": null,
  "id": "5vv2mz32z449k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.04247",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.04247",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vv2mz32z449k.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.04247",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.04247"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Weighted Fractional Commutator Estimate and Application to the Fractional Wave Equations with Time-dependent Damping",
  "title_text": "Weighted Fractional Commutator Estimate and Application to the Fractional Wave Equations with Time-dependent Damping",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/5vv2mz32z449k",
  "venue": "arXiv",
  "versions": []
}