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Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

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Abstract

This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form f(x)g(t) and f(x1, x2, t)h(x3), where g(t) and h(x3) are given and x = (x1, x2, x3) is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variables. For moving point sources radiating periodic signals, the data recorded at four receivers are prove sufficient to uniquely recover the orbit function. Simultaneous determination of embedded obstacles and source terms was verified in an inhomogeneous background medium using the observation data of infinite time period. Our approach depends heavily on the Laplace transform.

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Acknowledgments

The authors would like to thank Gen Nakamura for pointing the paper [24] and for helpful discussions.

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100193, China

    Guang-hui Hu

  2. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Yavar Kian

  3. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Yue Zhao

Authors
  1. Guang-hui Hu
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  2. Yavar Kian
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  3. Yue Zhao
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Corresponding author

Correspondence to Yue Zhao.

Additional information

The work of G. Hu is supported by the National Natural Science Foundation of China (No. 11671028) and the NSAF grant (No. U1930402) in the National Natural Science Foundation of China. The work of Y. Kian is supported by the French National Research Agency ANR (project MultiOnde) grant ANR-17-CE40-0029.

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Hu, Gh., Kian, Y. & Zhao, Y. Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains. Acta Math. Appl. Sin. Engl. Ser. 36, 134–150 (2020). https://doi.org/10.1007/s10255-020-0917-4

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  • DOI: https://doi.org/10.1007/s10255-020-0917-4

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