{
  "abstract": "We study the maximal existence time $T^{*}(\\varrho)$ of the solution of the semilinear wave equation $u_{tt}-Δu=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}$ in a bounded domain, with Dirichlet boundary condition and initial data $(\\varrho f,\\varrho g)$, where $1<q<p$ and the amplitude $\\varrho$ is large. For nontrivial $f$ and sufficiently large $\\varrho$, the concavity method gives $T^{*}(\\varrho)\\leq C\\varrho^{-(p-1)/2}$ for $q\\leq2p/(p+1)$ and $T^{*}(\\varrho)\\leq C\\varrho^{-(p-q)/q}$ for $q>2p/(p+1)$, whereas the energy method gives a lower bound of order $\\varrho^{1-p}$ only. We prove lower bounds with the same exponents as the upper ones, so that $T^{*}(\\varrho)\\asymp\\varrho^{-\\vartheta(p,q)}$ with $\\vartheta(p,q)=\\min\\{(p-1)/2,\\,(p-q)/q\\}$. The proof rests on a hyperbolic rescaling which converts the large amplitude into a dilation of the domain and a coefficient $\\varrho^{(q(p+1)-2p)/2}$ in front of the damping term, on a local existence theory in uniformly local energy norms whose existence time does not depend on that coefficient, and on a quantitative use of the dissipation when the coefficient is large. The threshold $2p/(p+1)$ is the value of $q$ at which the damping term is invariant under the rescaling. No attempt is made to optimize the constants: sharpness is meant throughout at the level of the exponent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10819",
  "author_total": 1,
  "authors": [
    {
      "name": "Firas Kaabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0mn1jpfvvwj7e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10819",
  "fulltext_url": null,
  "id": "0mn1jpfvvwj7e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10819",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7212321237",
  "pdf_url": "https://arxiv.org/pdf/2609.10819",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0mn1jpfvvwj7e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.10819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping",
  "title_text": "Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/0mn1jpfvvwj7e",
  "venue": "arXiv",
  "versions": []
}