{
  "abstract": "We study the dependence of parabolic solution operators on a finite signed measure describing the delay law. For a positive self-adjoint generator with compact inverse, a weighted dyadic spectral sum characterizes the norm of the semigroup-integrated memory perturbation. An exact realization identity transfers the matching lower estimate to two positive point delays in a fixed linear equation, using a common smooth, finite-spectral history. For the Dirichlet Laplacian on a nonempty bounded open set, the sharp worst-case modulus on a bounded ball of continuous $L^2$-valued histories is $d\\sqrt{\\log(e/d)}$, where $d$ is the bounded-Lipschitz distance between the delay measures. This is a rough-history endpoint result: logarithmic spatial regularity of order $γ>1/2$ restores resolution-uniform Lipschitz stability in the linear model. The critical order $γ=1/2$ retains a square-root double-logarithmic loss. More generally, a reciprocal-square summability criterion over occupied spectral bands gives the exact weighted threshold, including sparse spectra. Consequences include sharp finite-resolution Lipschitz constants, worst-case errors for prescribed midpoint quadrature of the memory measure, and semilinear upper estimates under Hilbert-space-valued local Lipschitz assumptions. The results distinguish sensitivity to the delay law from spatial approximation error and do not preclude uniform approximation of positive-time states on rough history balls.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.06262",
  "author_total": 1,
  "authors": [
    {
      "name": "Lennon J. Shikhman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/afgxnfhkcwrs1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.06262",
  "fulltext_url": null,
  "id": "afgxnfhkcwrs1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.06262",
  "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": "W7212007784",
  "pdf_url": "https://arxiv.org/pdf/2609.06262",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/afgxnfhkcwrs1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.06262",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.06262"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp spectral-scale stability for parabolic equations with measure-valued delay",
  "title_text": "Sharp spectral-scale stability for parabolic equations with measure-valued delay",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/afgxnfhkcwrs1",
  "venue": "arXiv",
  "versions": []
}