{
  "abstract": "We prove the global existence and uniqueness of strong solutions to a constrained version of the damped nonlinear wave equation $$ \\vartheta_{tt}+γ\\vartheta_t-Δ\\vartheta+|\\vartheta|^{p-2}\\vartheta=0 $$ on a smooth bounded domain $\\varOmega\\subset\\mathbb{R}^d$, where the evolution is projected onto the tangent space of the Hilbert manifold $$\\mathcal{M} = \\left\\{ \\vartheta\\in L^2(\\varOmega):\\|\\vartheta\\|_{L^2(\\varOmega)}=1 \\right\\}, $$ which is the unit sphere in $L^2(\\varOmega)$. We assume that $$p\\in[2,\\infty)\\ \\text{ for }\\ d=1,2, \\ \\text{ while }\\ 2\\leq p\\leq \\frac{2(d-1)}{d-2} \\ \\text{ for }\\ d\\geq3.$$ By employing the Faedo--Galerkin approximation method, together with suitable a priori estimates and compactness arguments, we establish the global well-posedness of the problem. In particular, we show that the Hilbert manifold $\\mathcal{M}$ is invariant under the flow, and hence the $L^2$-constraint is preserved throughout the evolution. Using the \\emph{Lusternik--Schnirelmann theory}, we show that the corresponding stationary problem possesses at least countably many stationary solutions. We further investigate the long-time behaviour of solutions and prove that, along a subsequence, every solution of the constrained problem converges to a stationary solution by invoking \\emph{Webb's theorem} and \\emph{Barbalat's lemma}. When the initial data are sufficiently close to the first eigenfunction of the associated stationary problem, we show that the unique strong solution converges in $H_0^1(\\varOmega)$ to the unique positive ground-state solution.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02842",
  "author_total": 2,
  "authors": [
    {
      "name": "Harsh Tiwari"
    },
    {
      "name": "Manil T. Mohan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1wv6ezpdam10c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02842",
  "fulltext_url": null,
  "id": "1wv6ezpdam10c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02842",
  "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",
      "math.AP"
    ],
    [
      "openalex",
      "2207|Stability and Controllability of Differential Equations"
    ]
  ],
  "openalex": "W7207792955",
  "pdf_url": "https://arxiv.org/pdf/2609.02842",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1wv6ezpdam10c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2609.02842",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02842"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint",
  "title_text": "Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/1wv6ezpdam10c",
  "venue": "arXiv",
  "versions": []
}