Sharp spectral-scale stability for parabolic equations with measure-valued delay
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 L2-valued histories is d√(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.
Related
- Concentration of measure-valued solutions for semilinear parabolic equations
- Sharp Wasserstein stability and small-delay thresholds for evolution equations with measure-valued principal-part delays
- Regularity Estimates for Quasilinear Elliptic Equations with Measure Data and Matrix-Valued Weight
- From Navier-Stokes to Measure-Valued Euler-Brinkman Equations: Homogenization in the Inviscid Limit
- Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States
BibTeX
@misc{shikhman2026sharp,
title = {{Sharp spectral-scale stability for parabolic equations with measure-valued delay}},
author = {Lennon J. Shikhman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06262},
eprint = {2609.06262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06262},
}
RIS
TY - UNPB
TI - Sharp spectral-scale stability for parabolic equations with measure-valued delay
AU - Lennon J. Shikhman
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.06262
UR - https://arxiv.org/abs/2609.06262
AB - 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√(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.
ER -