Vestnik КRAUNC. Fiz.-Mat. nauki. 2025. vol. 50. no. 1. P. 9 – 21. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-50-1-9-21
Research Article
Full text in Russian
MSC 35M10, 35M12

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Dedicated to the 70th anniversary of Professor R.R. Ashurov and the 70th anniversary of Professor R.R. Ashurov

On the smoothness of a semi-periodic boundary value problem for a three-dimensional equation of the second kind, second order mixed type in an unbounded domain

S. Z. Djamalov¹, B. K. Sipatdinova²^{\ast}

¹V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 100174, Tashkent, University street, 9, Uzbekistan
²Tashkent State Transport University, 1000167, str. Temiryulchilar, 1. Tashkent, Uzbekistan

Abstract. In the work of A.V.Bitsadze it is shown that the Dirichlet problem for a mixedtype equation is incorrect. The question naturally arises: is it possible to replace the conditions of the Dirichlet problem with other conditions covering the entire boundary, which ensure the correctness of the problem? For the first time such boundary value problems (non-local boundary value problems) for a mixed-type equation were proposed and studied in the works of F.I. Frankl. As problems for mixed type equations of the second kind in bounded domains, which are close in formulation to those under study, are investigated in the work of S. Dzhamalov. For mixed-type equations of the second kind of the second order in unbounded domains, semi-periodic boundary value problems in the three-dimensional case have been practically not investigated. In this paper, we investigate the uniqueness, existence and smoothness of a generalized solution to a semiperiodic boundary value problem for a mixed-type equation of the second kind, second order in an unbounded domain. In this paper, we prove the uniqueness of a generalized solution to the problem using the energy integral method. To prove the existence and smoothness of a generalized solution to the problem, the methods of “ε-regularization” and a priori estimates using the Fourier transform were used.

Key words: mixed-type equation of the second kind, semi-periodic boundary value problem, Fourier transform, anisotropic Sobolev space, energy integral, uniqueness of solution, “ε-regularization”methods, a priori estimates, existence and smoothness of a generalized solution.

Received: 31.03.2025; Revised: 14.04.2025; Accepted: 17.04.2025; First online: 18.04.2025

For citation. Djamalov S. Z., Sipatdinova B. K. On the smoothness of a semi-periodic boundary value problem for a three-dimensional equation of the second kind, second order mixed type in an unbounded domain. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 50: 1, 9-21. EDN: DBEEJG. https://doi.org/10.26117/2079-6641-2025-50-1-9-21.

Funding. The work was supported by a grant from the Ministry of Higher Education, Science and Innovation of the Republic of Uzbekistan No. F-FA-2021-424.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: sbiybinaz@mail.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Djamalov S. Z., Sipatdinova B. K., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

References

1. Bitsadze A. V. Ill-posedness of the Dirichlet problem for equations of mixed type. Reports of the USSR Academy of Sciences, 1953. V. 122, No 2, pp. 167–170. (In Russian).
2. Frankl F. I. Izbrannyye trudy po gazovoy dinamike [Selected works on gas dynamics]. Moscow. 1973. 711 p. (In Russian).
3. Terekhov A.N. Nelokal’nyye krayevyye zadachi dlya uravneniy peremennogo tipa [Nonlocal boundary value problems for equations of variable type]. Nonclassical equations of mathematical physics. Novosibirsk: IM SO AN SSSR, 1985. pp.148–158. (In Russian).
4. Glazatov S. N. Nonlocal Boundary Value Problems for Mixed-Type Equations in a Rectangle. Siberian Math. J., 1985. Vol. 26, No 6, pp. 162-164. (In Russian).
5. Dzhamalov S. Z. On well-posedness of a nonlocal boundary value problem with constant coefficients for a mixed type equation of the second kind of the second order in space. Mathematical notes of NEFU, 2017. No 4, pp. 17-28. (In Russian).
6. Dzhamalov S. Z. On the smoothness of a nonlocal boundary value problem for a multidimensional mixed-type equation of the second kind in space. Journal of the Middle Volga Mathematics Society, 2019. Vol.21, No 1, pp. 24-33. (In Russian).
7. Dzhamalov S.Z. Nelokal’nyye krayevyye i obratnyye zadachi dlya uravneniy smeshannogo tipa [Nonlocal boundary value and inverse problems for equations of mixed type]. Tashkent. 2021, 176 p. (In Russian).
8. Dzhamalov S.Z., Ashurov R.R., Turakulov Sh.Kh. On a nonlocal boundary value problem of periodic type for the three-dimensional Tricomi equation in an unbounded prismatic domain. Bulletin of the Institute of Mathematics. 4(3). 2021. pp. 52-59.
9. Vragov V.N. Krayevyye zadachi dlya neklassicheskikh uravneniy matematicheskoy fiziki [Boundary Value Problems for Nonclassical Equations of Mathematical Physics]. Novosibirsk, NGU, 1983. (In Russian).
10. Lyons J.L., Magenes E. Neodnorodnyye granichnyye zadachi i ikh prilozheniya [Inhomogeneous boundary value problems and their applications]. Moscow. Mir, 1971. (In Russian).
11. Hermander L. Lineynyye differentsial’nyye operatory s chastnymi proizvodnymi [Linear partial differential operators]. Moscow. Mir, 1965. (In Russian).
12. Nikolsky S.M. Priblizheniye funktsiy mnogikh peremennykh i teoremy vlozheniya [Approximation of functions of several variables and embedding theorems]. Moscow. Nauka, 1977. (In Russian).
13. Sobolev S.L. Nekotoryye primeneniya funktsional’nogo analiza v matematicheskoy fizike [Some applications of functional analysis in mathematical physics]. Moscow. Nauka. 1988. 335 p. (In Russian).
14. Ladyzhenskaya O.A. Krayevyye zadachi matematicheskoy fiziki [Boundary value problems of mathematical physics]. Moscow, 1973. (In Russian).
15. Karatopraklieva M. G. On a nonlocal boundary value problem for an equation of mixed type. Differential Equations. 1991. vol. 27, No 1, pp. 68-79. (In Russian).
16. Kozhanov A.I. Krayevyye zadachi dlya opredeleniya matematicheskoy fiziki nechetnogo poryadka [Boundary Value Problems for Odd-Order Equations of Mathematical Physics. Novosibirsk], NSU, 1990. (In Russian).

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