Abstract
In this paper, we study the well-posedness of a linear inverse problem for a mixed-type second-order multidimensional equation of the first kind. We prove the unique solvability of this problem in a certain function class with the help of “ε-regularization”, a priori estimates, and successive approximation methods.
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References
Bitsadze, A.V. “Ill-Posedness of the Dirichlet Problem for Equations of Mixed Type”, Dokl. Akad, Nauk SSSR 122 (2), 167–170 (1953) [in Russian].
Frankl, F.I., “Subsonic Flow about a Profile with a Supersonic Zone Terminated by a Direct Shock Wave”, Prikl. Matem. i Mekhan. 20 (2), 196–202 (1956) [in Russian].
Dzhamalov, S.Z. “A Nonlocal Boundary Value Problem with Constant Coefficients for the Multidimensional Second Order Equation of Mixed Type of the Second Kind”, Vestn. Samarsk. Gos. Tekhn. Un-ta. Ser. Fiz.-Matem. Nauki 21 (4), 1–14 (2017) [in Russian].
Kal’menov, T.Sh. “The Semiperiodic Dirichlet Problem for a Class of Equations of Mixed Type”, Differents. Uravn. 14 (3), 546–548 (1978).
Tsybikov, B.N. Well-Posedness of a Periodic Problem for a Multidimensional Equation of Mixed Type (in: Nonclassical partial differential equations, Novosibirsk, 201–206 (1986)).
Anikonov, Yu.E. Some Methods of Investigations of Multidimensional Inverse Problems for Differential Equations (Nauka, Novosibirsk, 1978) [in Russian].
Kabanikhin, S.I. Inverse and Ill-Posed Problems (Sib. Nauch. Izd-vo, Novosibirsk, 2009) [in Russian].
Lavrent’ev, M.M., Romanov, V.G., Vasil’ev, V.G., Multidimensional Inverse Problems for Differential Equations (Nauka, Novosibirsk, 1969) [in Russian].
Kozhanov, A.I., “Nonlinear Loaded Equations and Inverse Problems”, Zhurn. Vychisl. Matem. i Matem. Fiz. 44 (4), 694–716 (2004).
Sabitov, K.B., Sidorov, S.N. “Inverse Problem for Degenerate Parabolic-Hyperbolic Equation with Nonlocal Boundary Condition”, Izv. Vuz. Mat. 1, 46–59 (2015).
Sabitov, K.B., Martem’yanova, N.V., “Inverse Problem for Lavrent’ev-Bitsadze Equation Connected with the Search of Elements in Right-Hand Side”, Izv. Vuz. Mat. 2, 44–57 (2017).
Dzhamalov, S.Z. “The Linear Inverse Problem for the Mixed Type Equation of the Second Kind of the Second Order with Nonlocal Boundary Conditions in Three-Dimensional Space”, Uzb. Matem. Zhurn. 2, 58–65 (2017).
Dzhamalov, S.Z. “The Linear Inverse Problem for the Mixed Type Equation of the Second Kind of the Second Order with Nonlocal Boundary Conditions in Three-Dimensional Space”, Vestnik KRAUNC 17 (1), 7–13 (2017).
Nakhushev, A.M. “Loaded Equations and Their Applications”, Differents. Uravn. 19 (1), 86–94 (1983).
Trenogin, V.A. Functional Analysis (Nauka, Moscow, 1980) [in Russian].
Vragov, V.N. Boundary Value Problems for Nonclassic Equations of Mathematical Physics (NGU, Novosibirsk, 1983) [in Russian].
Ladyzhenskaya, O.A. Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973) [in Russian].
Acknowledgments
The authors are grateful to the academician of the Academy of Sciences of Republic of Uzbekistan Shavkat Arifzhanovich Alimov for the discussion of results obtained in this paper and for valuable advices, and to the reviewer for useful remarks.
Funding
This work was performed in accordance with the research plan OT-FA-88 and supported by the grant MRU-OT-1/2017.
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Russian Text © The Author(s), 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 6, pp. 11–22.
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Dzhamalov, S.Z., Ashurov, R.R. A Linear Inverse Problem for a Multidimensional Mixed-Type Second-Order Equation of the First-Kind. Russ Math. 63, 8–18 (2019). https://doi.org/10.3103/S1066369X19060021
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DOI: https://doi.org/10.3103/S1066369X19060021