Boundary problems for a special class of degenerate hyperbolic equations
https://doi.org/10.25587/2411-9326-2024-3-3-14
Abstract
We study the solvability of new boundary problems for a special class of degenerate second-order hyperbolic equations. These problems have two features. The first is the presence of two variables in the equation, each of which can be considered as a time variable. This means that problems with fundamentally different boundary conditions can be correct for these equations. The second feature is the presence of degeneracy. This means that the boundary problems formulation can change depending on the nature of degeneration.
For the problems under study,we prove theorems of existence and uniqueness for regular solutions are proven, i.e. solutions that have all generalized according to Sobolev derivatives included in the equation.
About the Authors
A. I. Kozhanov
Sobolev Institute of Mathematics
Russian Federation
Russian Federation
Alexandr I. Kozhanov
4 Koptyug Avenue, Novosibirsk 630090
N. R. Spiridonova
Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics
Russian Federation
Russian Federation
Naryia R. Spiridonova
48 Kulakovsky Street, Yakutsk 677000
References
1. Lin C. C., Reissner E., and Tsien H. S., “On two-dimensional nonsteady motion of a slender body in a compressible fluid,” J. Math. Phys., 27, No. 3, 220–231 (1948).
2. Mamontov Е. V., “On the equations of small disturbances in a non-stationary transonic gas flow [in Russian],” in: Non-Stationary Problems of Mechanics, No. 37, pp. 139–143, Inst. Gidrodin., Novosibirsk (1978).
3. Larkin N. A., “To the theory of the linearized Lin–Reissner–Tsien equation [in Russian],” in: Correct Boundary Value Problems for Non-Classical Equations of Mathematical Physics, pp. 126–131, Inst. Mat., Novosibirsk (1980).
4. Larkin N. A., “On the linearized equations of non-stationary gas dynamics [in Russian],” in: Non-Classical Equations and Equations of Mixed Type, pp. 107–118, Inst. Mat., Novosibirsk (1983).
5. Kozhanov A. I., “On the formulation and solvability of a boundary value problem for one class of equations not solved with respect to the time derivative [in Russian],” in: Boundary Value Problems for Non-Classical Equations of Mathematical Physics, pp. 84–98, Inst. Mat., Novosibirsk (1987).
6. Glazatov S. N., “A problem with data on a characteristic for a linearized equation of transonic gas dynamics,” Sib. Math. J., 37, No. 5, 898–906 (1996).
7. Glazatov S. N., “On solvability of a spatial periodic problem for the Lin–Reissner–Tsien equation of transonic gas dynamics,” Math. Notes, 87, No. 1–2, 130–134 (2010).
8. Varlamova G. A. and Kozhanov A. I., “Nonlocal problems with integral conditions for hyperbolic equations with two time variables [in Russian],” Mat. Zametki SVFU, 30, No. 3, 12–26 (2023).
9. Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc., Providence, RI (1991) (Transl. Math. Monogr.; vol. 90).
10. Yakubov S. Ya., Linear Differential-Operator Equations and Their Applications [in Russian], Elm, Baku (1985).
11. Kozhanov А. I., Composite Type Equations and Inverse Problems, VSP, Utrecht (1999).
12. Trenogin V. A., Functional Analysis [in Russian], Nauka, Moscow (1980).
Review
For citations:
Kozhanov A.I., Spiridonova N.R. Boundary problems for a special class of degenerate hyperbolic equations. Mathematical notes of NEFU. 2024;31(3):3-14. (In Russ.) https://doi.org/10.25587/2411-9326-2024-3-3-14
Views:
10
JATS XML
Email this article 