High frequency wave propagation for the viscoelastic wave equation with singular memory
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 m(s,x)=sp−1m(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≪1 yields the memory factor ε=h1−p. We construct exact solutions with full two-scale geometric-optics expansions in powers hk+(1−p)ℓ. Memory modifies the propagation geometry through the instantaneous modulus σ+ε∫0∞m(s,·) d s, while the kernel singularity contributes Cp=Γ(p)eiπp/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 Ck norms. Exterior observations for all incident directions and 0<h≪1 uniquely recover σm and the full temporal jet of m at s=0, which determine the expansion modulo O(h∞). 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 ε. These estimates justify the geometric-optics construction.
Related
- Note on the Exponential Stability of the Viscoelastic Wave Equation
- Blow-up at arbitrary energy levels for a strongly damped viscoelastic wave equation with a variable-exponent logarithmic source
- Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
- Well-posedness and Stability of Thermoelastic Suspension Bridges with Cattaneo Heat Conduction: The Role of Viscoelastic Memory
- A Viscoelastic Wave Equation with Nonlinear Damping and Supercritical Variable-Exponent Logarithmic Sources
BibTeX
@misc{grebnev2026high,
title = {{High frequency wave propagation for the viscoelastic wave equation with singular memory}},
author = {Haim Grebnev and Plamen Stefanov},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30138},
eprint = {2608.30138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30138},
}
RIS
TY - UNPB
TI - High frequency wave propagation for the viscoelastic wave equation with singular memory
AU - Haim Grebnev
AU - Plamen Stefanov
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30138
UR - https://arxiv.org/abs/2608.30138
AB - 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 m(s,x)=s^p−1m(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≪1 yields the memory factor ε=h^1−p. We construct exact solutions with full two-scale geometric-optics expansions in powers h^k+(1−p)ℓ. Memory modifies the propagation geometry through the instantaneous modulus σ+ε∫_0^∞m(s,·) 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≪1 uniquely recover σ_m and the full temporal jet of m at s=0, which determine the expansion modulo O(h^∞). 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 ε. These estimates justify the geometric-optics construction.
ER -