AI 中文总结
本文研究黎曼流形上带时间非局部衰减的波动方程，利用边界测量恢复时间相关低阶系数，并证明在任意小或不相交时间区间上测量的可行性，结合多种数学工具。
AI 中文摘要
本文研究了在黎曼流形上具有时间非局部衰减的波动方程中，利用边界测量来确定时间相关的一阶和零阶系数的反问题。该问题源于成像和粘弹性等应用，在这些应用中此类方程自然出现，我们旨在通过恢复这些系数来刻画底层介质的性质。我们的主要目标是利用由非局部衰减引起的解的记忆性质，从任意小时间区间内收集的边界测量中恢复一般类的低阶系数。作为我们分析的副产品，我们还证明了利用不相交时间区间上的测量来恢复时间相关系数的结果。虽然这些数据限制对于经典波动方程是不可用的，且通常是不可能的，但我们证明，在标准假设下，由于时间非局部衰减的存在，这些限制变得可行。我们的分析结合了微分几何、具有非局部项的偏微分方程理论、复分析以及非局部算子理论的技术。
英文摘要
In this article, we investigate the inverse problem of determining time-dependent first- and zeroth-order coefficients for a wave equation with nonlocal-in-time attenuation on a Riemannian manifold, using boundary measurements. This problem is motivated by applications in imaging and viscoelasticity, where such equations arise naturally, and we aim to characterize the properties of the underlying medium by recovering these coefficients. Our main objective is to exploit the memory properties of solutions induced by nonlocal attenuation, establishing the recovery of a general class of lower-order coefficients from boundary measurements collected over an arbitrarily small time interval. As a byproduct of our analysis, we also prove recovery results for time-dependent coefficients using measurements supported on disjoint time intervals. While such data restrictions are unavailable, and generally impossible, for the classical wave equation, we demonstrate that they become feasible under standard assumptions due to the presence of nonlocal-in-time attenuation. Our analysis combines techniques from differential geometry, the theory of partial differential equations with nonlocal terms, complex analysis, and the theory of nonlocal operators.
