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 $\mathfrak m(s,x)=s^{p-1}m(s,x)$, $0<p<1$; 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 $\sigma+\varepsilon\int_0^\infty\mathfrak m(s,\cdot)\,d s$, while the kernel singularity contributes $C_p=\Gamma(p)e^{i\pi p/2}$ 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 $\sigma_{\mathfrak m}$ 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.