Nonlocal problem for a class of third-order equations
https://doi.org/10.25587/SVFU.2023.45.27.001
Abstract
This article considers a nonlocal problem in a cylindrical domain for the third-order mixed-composite type equation of the form
uttt −µ(x1) ∂
∂x1
�u −a(x, t)�u = f(x, t),
where x1µ(x1) > 0 for x1 ̸= 0, µ(0) = 0, x = (x1, x2, . . . , xn) ∈Rn.
Using the Galerkin method, it is proved that this nonlocal problem, under certain conditions on the coefficients and the right side of the equation, has a unique solution in Sobolev spaces. The proof is based on the Galerkin method with the choice of a special basis and a priori estimates. New theorems are also proved regarding the existence and uniqueness of the solution of the nonlocal problem, which allow expanding the range of solvable problems in the theory of boundary value problems for nonclassical equations of mathematical physics.
References
1. Bitsadze A. V. and Salakhitdinov M. S., “On the theory of equations of mixed-composite type [in Russian],” Sib. Mat. Zh., 2, No. 1, 7–19 (1961).
2. Bitsadze A. V., “On equations of mixed-composite type,” in: Some Problems of Mathematics and Mechanics [in Russian], pp. 47–49, Izdat. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1961).
3. Vragov V. N., “On one equation of mixed-composite type [in Russian],” Differ. Uravn., 9, No. 1, 169–171 (1973).
4. Salakhitdinov M. S., Equations of Mixed-Composite Type [in Russian], Fan, Tashkent (1974).
5. Djuraev T. D. and Rakhmanov U., “On well-posed boundary value problems for equations of mixed-composite type [in Russian],” Differ. Uravn., 9, No. 1, 32–40 (1973).
6. Bobylova L. A. and Smirnov M. M., “On a boundary value problem for an equation of mixedcomposite type of the 4th order [in Russian],” Izv. VUZov, Mat., No. 5, 15–21 (1972).
7. Pyatkov S. G., “On one equation of composite type [in Russian],” Differ. Uravn., 16, No. 1, 117–123 (1980).
8. Abulov M. A., “Boundary value problem for a mixed-composite type of the third order [in Russian],” Novosib. Gos. Univ., Novosibirsk (1989). 5 p. (Dep. VINITI, No. 1436, 89).
9. Egorov I. E. and Fedorov V. E., Non-Classical Equations of Mathematical Physics of High Order [in Russian], Nauka, Novosibirsk (1995).
10. Mamedov I. G., Solution of Multidimensional Local and Nonlocal Boundary Value Problems for High Order Hyperbolic Equations with Nonsmooth Coefficients and Their Application to Optimal Control Problems [in Russian], Baku (2015).
11. Takhirov J. O., Non-Classical Nonlinear Problems and Problems with a Free Boundary [in Russian], Tashkent (2014).
12. Balkizov Zh. A., “A non-local boundary value problem for an equation of parabolic-hyperbolic type of the third order with degeneration of type and order in the region of its hyperbolicity [in Russian],” Itogi Nauki i Tekhniki, Sovrem. Mat. Pril., 149, 14–24 (2018).
13. Kozhanov A. I., “Time-nonlocal boundary value problem for linear parabolic equations,” Sib. J. Ind. Math., 7, No. 1, 51–60 (2004).
14. Abdurakhmanov A. M. and Kozhanov A. I., “Problem with a nonlocal boundary condition for a class of odd-order equations [in Russian],” Izv. VUZov, Mat., No. 5, 3–12 (2007).
15. Pyatkov S. G., “On some classes of nonlocal boundary value problems for singular parabolic equations [in Russian],” Mat. Zamet., 106, No. 4, 578–594 (2019).
16. Alimov Sh. A. and Pulatov A. K., “On a nonlocal Bitsadze–Samarskii problem [in Russian],” Izv. Akad. Nauk Uzbek. SSR, Fiz.-Mat. Nauki, No. 1, 8–11 (1986).
17. Zikirov O. S., “On the solvability of a nonlocal problem for a third-order hyperbolic equation [in Russian],” Sib. Zh. Chist. Prikl. Mat., 16, No. 2, 16–26 (2016).
18. Artyushin A. N., “On the regular solvability of a boundary value problem for a third-order equation with a changing direction of evolution in weighted Sobolev spaces [in Russian],” Sib. Electron. Math. Rep., 16, 2003–2012 (2019).
19. Abulov M. O., “Nonlocal problem for one class of third order equations [in Russian],” Karshi, KarDU Khabarl., 3, No. 1, 12–16 (2022).
20. Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics [in Russian], Izdat. Sib. Otdel. Akad. Nauk SSSR, Novosibirsk (1962).
Review
For citations:
Abulov M.O. Nonlocal problem for a class of third-order equations. Mathematical notes of NEFU. 2023;30(3):3-11. (In Russ.) https://doi.org/10.25587/SVFU.2023.45.27.001
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