# Vestnik КRAUNC. Fiz.-Mat. Nauki. 2023. vol. 42. no. 1. P. 58-68. ISSN 2079-6641

**MATHEMATICS **

https://doi.org/10.26117/2079-6641-2023-42-1-58-68

Research Article

Full text in Russian

MSC 35B10

**On a nonlocal boundary value problem of periodic type for the three-dimensional mixed-type equations of the second kind in an infinite parallelepiped**

**S. Z. Dzhamalov^*, B. K. Sipatdinova^***

Institute of Mathematics named after V. I. Romanovskiy, Academy of Sciences of the Republic of Uzbekistan, 100174, Tashkent, Universitet str., 4b, Uzbekistan

Abstract. As is known, A.V. Bitsadze in his studies pointed out that the Dirichlet problem for a mixed-type equation, in particular for a degenerate hyperbolic-parabolic equation, is ill-posed. The question naturally arises: is it possible to replace the conditions of the Dirichlet problem with other conditions covering the entire boundary, which will ensure the well-posedness of the problem? For the first time, such boundary value problems (nonlocal boundary value problems) for a mixed-type equation were proposed and studied in the works of F.I. Frankl when solving the gas-dynamic problem of subsonic flow around airfoils with a supersonic zone ending in a direct shock wave. Problems close in formulation to a mixed-type equation of the second order were considered in the studies by A.N. Terekhov, S.N. Glazatov, M.G. Karatopraklieva and S.Z. Dzhamalov. In these papers, nonlocal boundary value problems in bounded domains are studied for a mixed-type equation of the second kind of the second order. Such problems for a mixed-type equation of the first kind in the three-dimensional case (in particular, for the Tricomi equation) in unbounded domains are studied in the works of S.Z. Dzhamalov and H. Turakulov. For mixed-type equations of the second kind in unbounded domains, nonlocal boundary value problems in the multidimensional case are practically not studied. In this article, nonlocal boundary value problem of periodic type for a mixed-type equation of the second kind of the second order, is formulated and studied in an unbounded parallelepiped. To prove the uniqueness of the generalized solution, the method of energy integrals is used. To prove the existence of a generalized solution, the Fourier transforms is used and as a result, a new problem is obtained on the plane. And for the solvability of this problem, the methods of “*\varepsilon*-regularization”and a priori estimates are used. The uniqueness, existence, and smoothness of a generalized solution of a nonlocal boundary value problem of periodic type for a three-dimensional mixed-type equation of the second kind of the second order are proved using above-mentioned methods and Parseval equality.

*Key words: mixed-type equation of the second kind, nonlocal boundary value problem, Fourier transform, methods of “\varepsilon-regularization”and a priori estimates.*

Received: 11.08.2022; Revised: 05.12.2022; Accepted: 24.03.2023; First online: 15.04.2023

**For citation.** Dzhamalov S. Z., Sipatdinova B. K. On a nonlocal boundary value problem of periodic type for the

three-dimensional mixed-type equations of the second kind in an infinite parallelepiped. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 42: 1, 58-68. EDN: GMDAQU. https://doi.org/10.26117/2079-6641-2023-42-1-58-68.

**Funding.** The authors acknowledge financial support from the Ministry of Innovative Development of the Republic of

Uzbekistan, Grant no. F-FA-2021-424.

**Competing interests.** The authors declare that 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.

^**Correspondence: E-mail: siroj63@mail.ru, sbiybinaz@mail.ru*

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

© Dzhamalov S. Z., Sipatdinova B. K., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

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** Information about authors **

**Dzhamalov Sirojiddin Zuxriddinovich** – D. Sci. (Phys. & Math.), Associate Professor, Chief Researcher of the V.I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan, https://orcid.org/0000-0002-3925-5129.

**Sipatdinova Biybinaz Kenesbayevna** – Ph. D. (Phys. & Math.) student of the V.I. Romanovsky Institute of Mathematics of the Academy of Sciences of the Republic of Uzbekistan, Tashkent, Uzbekistan, https://orcid.org/0000-0002-7833-6992.