未知终端时刻观测下的次扩散反问题
Inverse Problems for Subdiffusion from Observation at an Unknown Terminal Time
摘要：
恢复空间相关参数的逆问题，例如，近年来，在终端观测的次扩散模型中，空间相关源或势系数的初始条件得到了广泛的研究。然而，所有现有的研究都假设人们进行观察的终端时间是确切知道的。在这项工作中，我们提出了三个典型的反问题，例如，反向问题，反源和反势问题，从终端观测在一个未知的时间。问题的次扩散性质表明，可以同时确定终端的时间和空间相关的参数。分析是基于显式的解决方案表示，Mittag-Leffler函数的渐近行为，和温和的正则性条件的问题数据。此外，我们提出了几个一维和二维的数值实验来说明该方法的可行性。
英文摘要：
Inverse problems of recovering space-dependent parameters, e.g., initial condition, space-dependent source or potential coefficient, in a subdiffusion model from the terminal observation have been extensively studied in recent years. However, all existing studies have assumed that the terminal time at which one takes the observation is exactly known. In this work, we present uniqueness and stability results for three canonical inverse problems, e.g., backward problem, inverse source and inverse potential problems, from the terminal observation at an unknown time. The subdiffusive nature of the problem indicates that one can simultaneously determine the terminal time and space-dependent parameter. The analysis is based on explicit solution representations, asymptotic behavior of the Mittag-Leffler function, and mild regularity conditions on the problem data. Further, we present several one- and two-dimensional numerical experiments to illustrate the feasibility of the approach.
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