{
  "abstract": "We study high-frequency propagation for a viscoelastic wave equation with spatially dependent hereditary memory in relative-history form. The kernel may have the integrable singularity $\\mathfrak m(s,x)=s^{p-1}m(s,x)$, $0<p<1$; the regular case $p=1$ is included. Using the half-amplitude propagation distance and corresponding travel time as units, the wavelength $h\\ll1$ yields the memory factor $\\varepsilon=h^{1-p}$. We construct exact solutions with full two-scale geometric-optics expansions in powers $h^{k+(1-p)\\ell}$. Memory modifies the propagation geometry through the instantaneous modulus $σ+\\varepsilon\\int_0^\\infty\\mathfrak m(s,\\cdot)\\,d s$, while the kernel singularity contributes $C_p=Γ(p)e^{iπp/2}$ to the leading transport equation. For $0<p<1$, this produces fractional scales, frequency-dependent attenuation, and a dispersive phase correction; for $p=1$, the fractional hierarchy disappears, attenuation is frequency independent, and the transport phase correction vanishes. We also derive a local damped wave equation whose incoming high-frequency solutions approximate the hereditary solutions with $O(h)$ error in semiclassical $C^k$ norms. Exterior observations for all incident directions and $0<h\\ll1$ uniquely recover $σ_{\\mathfrak m}$ and the full temporal jet of $m$ at $s=0$, which determine the expansion modulo $O(h^\\infty)$. Finally, a contraction-semigroup argument gives well-posedness and arbitrary finite-order Sobolev regularity for spatially dependent weakly singular kernels and prescribed full prehistory, with explicit compatibility conditions and estimates uniform in $\\varepsilon$. These estimates justify the geometric-optics construction.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30138",
  "author_total": 2,
  "authors": [
    {
      "name": "Haim Grebnev"
    },
    {
      "name": "Plamen Stefanov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/72bpzeb7fh5kz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30138",
  "fulltext_url": null,
  "id": "72bpzeb7fh5kz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30138",
  "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": "W7204864560",
  "pdf_url": "https://arxiv.org/pdf/2608.30138",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/72bpzeb7fh5kz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30138",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30138"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "High frequency wave propagation for the viscoelastic wave equation with singular memory",
  "title_text": "High frequency wave propagation for the viscoelastic wave equation with singular memory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/72bpzeb7fh5kz",
  "venue": "arXiv",
  "versions": []
}