Nonlocal problems with integral conditions for hyperbolic equations with two time variables
https://doi.org/10.25587/SVFU.2023.99.74.002
Abstract
The work is devoted to the study of solvability of boundary value problems with nonlocal conditions of integral form for the differential equations
uxt −auxx + c(x, t)u = f(x, t),
in which x ∈�= (0, 1), t ∈(0, T), 0 < T < +∞, a ∈R, and c(x, t) and f(x, t) are
known functions.
The peculiarity of these equations is that any of variables t and x can be considered a temporary variable, and in accordance with this, for these equations, for- mulations of boundary value problems with different carriers of boundary conditions can be proposed. For the problems under study, the work proves existence and uniqueness theorems for regular solutions; namely, solutions that have all derivatives generalized according to S. L. Sobolev and included in the equation.
About the Authors
G. A. Varlamova
Ammosov North-Eastern Federal University, Mirny Polytechnic Institute
Russian Federation
Russian Federation
Galina A. Varlamova
5/1 Tikhonov Street, Mirny 678175
A. I. Kozhanov
Sobolev Institute of Mathematics
Russian Federation
Russian Federation
Aleksandr I. Kozhanov
4 Koptyug Avenue, 630090 Novosibirsk
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. Larkin N. A., “On the theory of the linearized Lin–Reissner–Jian equation,” in: Correct Boundary Value Problems for Non-Classical Equations of Mathematical Physics, pp. 126–131, Inst. Math., Novosibirsk (1980).
3. Larkin N. A., “On the linearized equation of non-stationary gas dynamics,” in: Non-Classical Equations and Equations of Mixed Type, pp. 107–118, Inst. Math., Novosibirsk (1983).
4. Kozhanov A. I., “On the formulation and solvability of a boundary value problem for one class of equations not resolved with respect to the time derivative,” in: Boundary Value Problems for Non-Classical Equations of Mathematical Physics, pp. 84–98, Inst. Math., Novosibirsk (1987).
5. Glazatov S. N., “Problem with data on the characteristic for linearized equations of transonic gas dynamics,” Sib. Math. J., 37, No. 5, 1019–1029 (1996).
6. Glazatov S. N., “On the solvability of a spatially periodic problem for the Lin–Reissner–Tsien equation of transonic gas dynamics,” Math. Notes, 87, No. 1, 137–140 (2010).
7. Kozhanov A. I. and Pulkina L. S., “On the solvability of a boundary value problem with nonlocal boundary condition of integral form for multidimensional hyperbolic equations,” Differ. Equ., 42, No. 9, 1166–1179 (2006).
8. Kozhanov A. I. and Pulkina L. S., “On the solvability of some boundary conditions problems with displacement for linear hyperbolic equations,” Math. J. (Kazakhstan), 9, No. 2, 78–92 (2009).
9. Kozhanov A. I., “On the solvability of boundary value problems with nonlocal and integral conditions for parabolic equations,” Nonlinear Boundary Probl., 20, 54–76 (2010).
10. Pulkina L. S., Problems with Nonclassical Conditions for Hyperbolic Equations, Izdat. Samar. Univ., Samara (2012).
11. Kozhanov A. I., “Nonlocal problems with integral conditions for elliptic equations,” Complex Variables Elliptic Equ., 64, No. 5, 741–752 (2019).
12. Artyushin A. N., “Volterra’s integral and evolutionary equations equations with integral condition,” Differ. Equ., 53, No. 7, 867–881 (2017).
13. Soldatov A. P. and Shkhanukov M. Kh., “Boundary value problems with a general nonlocal A. A. Samarsky’s condition for pseudoparabolic equations of high order,” Dokl. Akad. Nauk SSSR, 297, No. 3, 547–552 (1987).
14. Yusubov Sh. Sh. and Mamedova J. J., “On the solvability of the equation with dominant mixed derivative with integral boundary conditions,” Bull. Baku State Univ., Ser. Math., No 3, 57–62 (2012).
15. Popov N. S., “Solvability of a boundary value problem for a pseudohyperbolic equations with nonlocal integral conditions,” Differ. Equ., 51, No. 3, 359–372 (2015).
16. Popov N. S. and Kozhanov A. I., “On the solvability of some problems with displacement for pseudoparabolic equations,” Vestn. Novosib. Gos. Univ., Ser. Mat., Mekh. Inform., 10, No. 3, 63–75 (2010).
17. Popov N. S., “On the solvability of spatially nonlocal boundary conditions problems for onedimensional pseudoparabolic and pseudohyperbolic equations,” Vestn. Samar. Gos. Univ., 125, No. 3, 29–42 (2015).
18. Lukina G. A., “Boundary value problems with integral boundary conditions for the linearized Korteweg-de Vries equation,” Vestn. Yuzh.-Ural. Gos. Univ., Ser. Mat. Model., Program., 8, No. 17, 53–62 (2011).
19. Kozhanov A. I. and Lukina G. A., “Nonlocal boundary value problems with partially integral conditions for degenerate differential equations with multiple characteristics,” Sib. J. Pure Appl. Math., 17, No. 3, 37–51 (2017).
20. Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc., Providence, RI (1991) (Transl. Math. Monogr.; vol. 90).
21. Ladyzhenskaya O. A., Solonnikov V. A., and Uraltseva N. N., Linear and Quasilinear Parabolic Equations [in Russian], Nauka, Moscow (1967).
22. Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutcher Verl. Wiss., Berlin (1978).
23. Yakubov S. Ya., Linear Operator-Differential Equations and Their Applications [in Russian], Ehlm, Baku (1985).
24. Kozhanov A. I., Composite Type Equations and Inverse Problems, VSP, Utrecht (1999).
25. Larkin N. A., “Existence theorems for quasilinear pseudohyperbolic equations,” Sov. Math., Dokl., 26, 260–263 (1982).
26. Larkin N. A., Novikov V. A., and Yanenko N. N., Nonlinear Equations of Variable Type [in Russian], Nauka, Novisibirsk (1983).
27. Liu Y. and Li H., “H1-Galerkin mixed finite element methods for pseudo-hyperbolic equations,” Appl. Math. Comput., 212, No. 2, 446–457 (2009).
28. Khudaverdiyev K. I. and Veliyev A. A., Investigation of a One-Dimensional Mixed Problem for a Class of Pseudohyperbolic Equations of Third Order with Non-Linear Operator Right-Hand Side [in Russian], Chashyogly, Baku (2010).
29. Demidenko G. V. and Uspenskii S. V., Equations and Systems Not Resolved with Respect to the Highest Derivative [in Russian], Nauch. Kniga, Novosibirsk (1998).
30. Trenogin V. A., Functional Analysis, Nauka, Moscow (1980).
Review
For citations:
Varlamova G.A., Kozhanov A.I. Nonlocal problems with integral conditions for hyperbolic equations with two time variables. Mathematical notes of NEFU. 2023;30(3):12-26. (In Russ.) https://doi.org/10.25587/SVFU.2023.99.74.002
Views:
8
JATS XML
Email this article 