We consider an inverse problem of reconstructing a degeneracy point in the diffusion coefficient in a one-dimensional parabolic equation by measuring the normal derivative on one side of the domain boundary. We analyze the sensitivity of the inverse problem to the initial data. We give sufficient conditions on the initial data for uniqueness and stability for the one-point measurement and show some examples of positive and negative results. On the other hand, we present more general uniqueness results, also for the identification of an initial data by measurements distributed over time. The proofs are based on an explicit form of the solution by means of Bessel functions of the first type. Finally, the theoretical results are supported by numerical experiments.

中文翻译：

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抛物方程简并电导率区域的重构

我们考虑通过测量域边界一侧的法向导数来重建一维抛物线方程中扩散系数简并点的逆问题。我们分析反问题对初始数据的敏感性。我们对初始数据给出了单点测量的唯一性和稳定性的充分条件，并展示了一些正面和负面结果的示例。另一方面，我们提出了更一般的唯一性结果，也用于通过随时间分布的测量来识别初始数据。证明基于通过第一类型贝塞尔函数的解的显式形式。最后，理论结果得到数值实验的支持。