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The uniqueness of inverse problems for a fractional equation with a single measurement

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Abstract

This article is concerned with an inverse problem on simultaneously determining some unknown coefficients and/or an order of derivative in a multidimensional time-fractional evolution equation either in a Euclidean domain or on a Riemannian manifold. Based on a special choice of the Dirichlet boundary input, we prove the unique recovery of at most two out of four \(\varvec{x}\)-dependent coefficients (possibly with an extra unknown fractional order) by a single measurement of the partial Neumann boundary output. Especially, both a vector-valued velocity field of a convection term and a density can also be uniquely determined. The key ingredient turns out to be the time-analyticity of the decomposed solution, which enables the construction of Dirichlet-to-Neumann maps in the frequency domain and thus the application of inverse spectral results.

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Acknowledgements

The authors thank the anonymous referees for their valuable comments. This work is partly supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP15H05740. The first author is partially supported by the French National Research Agency ANR (project MultiOnde) grant ANR-17-CE40-0029. The second author is supported by National Natural Science Foundation of China (NSFC) (No. 11801326). The third author is supported by JSPS KAKENHI Grant Number JP20K14355. The fourth author is partly supported by NSFC (Nos. 11771270, 91730303). This work was prepared with the support of the “RUDN University Program 5-100”.

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Authors and Affiliations

  1. Aix-Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France

    Yavar Kian

  2. School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong, 255049, China

    Zhiyuan Li

  3. Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, N12W7, Kita-Ward, Sapporo, 060-0812, Japan

    Yikan Liu

  4. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Masahiro Yamamoto

  5. Honorary Member of Academy of Romanian Scientists, Splaiul Independentei Street, No. 54, 050094, Bucharest, Romania

    Masahiro Yamamoto

  6. Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, Moscow, 117198, Russian Federation

    Masahiro Yamamoto

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  1. Yavar Kian
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  2. Zhiyuan Li
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  4. Masahiro Yamamoto
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Correspondence to Yikan Liu.

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Communicated by Y. Giga.

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Kian, Y., Li, Z., Liu, Y. et al. The uniqueness of inverse problems for a fractional equation with a single measurement. Math. Ann. 380, 1465–1495 (2021). https://doi.org/10.1007/s00208-020-02027-z

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  • DOI: https://doi.org/10.1007/s00208-020-02027-z

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