{
  "abstract": "We study the large-time behavior of the $L^2$-norm of solutions to the free wave equation on $\\mathbb{R}^N$ and, more generally, their homogeneous Sobolev norms of negative order. We refine and extend the results of R. Ikehata [J. Hyperbolic Differ. Equ. 20 (2023)] who identified and quantified the infinite-time $L^2$-growth of free waves in dimensions $N=1,2$. For finite-energy data with $u_1\\in L^1(\\mathbb{R}^N)$, we derive exact first-order asymptotics: when $P=\\int_{\\mathbb{R}^N}u_1(x)\\,dx\\neq0$, the squared $L^2$-norm grows linearly for $N=1$ and logarithmically for $N=2$, with a leading coefficient depending only on $P$. In particular, no weighted moment assumption on $u_1$ is required; $u_1\\in L^1\\cap L^2$ suffices. More generally, we prove that the $L^2$-norm is bounded, and hence convergent, if and only if $u_1\\in\\dot H^{-1}(\\mathbb{R}^N)$. We further establish first-order asymptotic profiles in negative-order homogeneous Sobolev spaces, covering polynomial growth, logarithmic growth, and convergence to a finite limit according to the Sobolev index. These results reveal a unified low-frequency mechanism underlying the low-dimensional $L^2$-growth. We also derive second-order asymptotic expansions in dimension one and at the critical Sobolev index in dimensions $N\\ge2$. Finally, in dimension two, the exact $L^2$-asymptotics yield a sharper local energy decay estimate with an explicit upper asymptotic constant independent of the localization radius.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15086",
  "author_total": 2,
  "authors": [
    {
      "name": "Vishvesh Kumar"
    },
    {
      "name": "Mohamed Majdoub"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dv0aqhg6e8ty4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15086",
  "fulltext_url": null,
  "id": "dv0aqhg6e8ty4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15086",
  "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|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7221825729",
  "pdf_url": "https://arxiv.org/pdf/2610.15086",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dv0aqhg6e8ty4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15086",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15086"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Exact $L^2$-growth profiles for the free wave equation: Sharp constants, critical regularity, and local energy decay",
  "title_text": "Exact L^2-growth profiles for the free wave equation: Sharp constants, critical regularity, and local energy decay",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/dv0aqhg6e8ty4",
  "venue": "arXiv",
  "versions": []
}