{
  "abstract": "We study the Cauchy problem $$ v_{tt}-v_{xx}=\\abs{v_t+v_x}^{m}\\abs{v_t}^{n}, \\qquad v(x,0)=ηφ(x),\\quad v_t(x,0)=ηψ(x), $$ on $\\R$, with compactly supported profiles $φ\\in C_0^2(\\R)$, $ψ\\in C_0^1(\\R)$, exponents $m,n>1$, and amplitude $η>0$. We show that the sign of $P_0=ψ+φ'$ decides the behaviour of small solutions, whatever the sign of $ψ-φ'$. If $P_0$ is positive at some point, the lifespan $T(η)$ obeys explicit two-sided bounds of order $η^{-(m+n-1)}$ for every $η>0$, and $$ \\lim_{η\\to0^+}η^{m+n-1}T(η)=\\frac{2^{n}}{(m+n-1)\\,(\\max_{\\R}P_0)^{m+n-1}} . $$ If $P_0\\le0$, the solution is global for every amplitude below an explicit threshold. If moreover $P_0\\not\\equiv0$, the component $v_t+v_x$ decays at the universal rate $\\bigl(2^{n}/((m+n-1)t)\\bigr)^{1/(m+n-1)}$, and the gradient of the solution converges uniformly to that of a free wave travelling to the right; if $P_0\\equiv0$, the solution is itself such a travelling wave. The analysis rests on a localization property: the zero set of $v_t+v_x$ is invariant along its own characteristics, so that the nonlinear source stays in a slab of fixed width moving with speed one, and each characteristic of the other family is forced only during a bounded time. This yields the global existence, the long-time behaviour and the exact value of the limit. The results extend to the endpoint exponents $m,n\\ge1$ and to sources $f(v_t+v_x)g(v_t)$, for which $T(η)$ is asymptotic to an Osgood-type integral.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28292",
  "author_total": 1,
  "authors": [
    {
      "name": "Firas Kaabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f0tyxr3m82szq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28292",
  "fulltext_url": null,
  "id": "f0tyxr3m82szq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28292",
  "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",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7204807769",
  "pdf_url": "https://arxiv.org/pdf/2608.28292",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f0tyxr3m82szq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2608.28292",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28292"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Characteristic localization, sharp lifespan asymptotics and global dynamics for a one-dimensional derivative wave equation",
  "title_text": "Characteristic localization, sharp lifespan asymptotics and global dynamics for a one-dimensional derivative wave equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/f0tyxr3m82szq",
  "venue": "arXiv",
  "versions": []
}