Single-Measurement Hölder Stability for a Spatial Coefficient in a Wave Equation with Principal Memory

Abstract

We study the recovery of a spatial coefficient in the principal memory flux of a three-dimensional wave equation with the prescribed exponential relaxation kernel $βe^{-βt}$, where $β>0$ is known. From a single finite-time lateral Cauchy measurement generated by one fixed, coefficient-independent initial displacement that is affine near the unknown region, with zero initial velocity, we establish global uniqueness and conditional $L^2$-Hölder stability. Two difficulties arise in this setting. The affine initial displacement makes the initial acceleration vanish in the unknown region, while the coefficient-difference equation contains a divergence-form source whose direct $L^2$ estimate loses a spatial derivative. We establish an acoustic Carleman estimate that controls the vector source and its time derivative without spatially differentiating the source. The affine condition identifies the relevant initial trace with a directional derivative of the coefficient difference, and the quadratic temporal vanishing of the source permits absorption in the weighted estimate. A transport estimate then yields stability uniformly over a bounded, uniformly coercive $W^{5,\infty}$ coefficient class, without an ordering condition or a smallness assumption near a reference coefficient.