Geodesic
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. Fiziko-matematičeskie nauki, Tome 42 (2023) no. 1, pp. 58-68 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 "$\epsilon$-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.
Keywords: mixed-type equation of the second kind, nonlocal boundary value problem, methods of "$\epsilon$-regularization"and a priori estimates.
Mots-clés : Fourier transform 
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S. Z. Djamalov; B. K. Sipatdinova. 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. Fiziko-matematičeskie nauki, Tome 42 (2023) no. 1, pp. 58-68. http://geodesic.mathdoc.fr/item/VKAM_2023_42_1_a3/

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