Geodesic
A nonlocal problem for a hyperbolic equation with a dominant mixed derivative
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 4, pp. 25-35 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this article, we consider the Goursat problem with nonlocal integral conditions for a hyperbolic equation with a dominant mixed derivative. Research methods of solvability of classical boundary value problems for partial differential equations cannot be applied without serious modifications. The choice of a research method of solvability of a nonlocal problem depends on the form of the integral condition. In the process of developing methods that are effective for nonlocal problems, integral conditions of various types were identified [1]. The solvability of the nonlocal Goursat problem with integral conditions of the first kind for a general equation with dominant mixed derivative of the second order was investigated in [2]. In our problem, the integral conditions are nonlocal conditions of the second kind, therefore, to investigate the solvability of the problem, we propose another method, which consists in reducing the stated nonlocal problem to the classical Goursat problem, but for a loaded equation. In this article, we obtain conditions that guarantee the existence of a unique solution of the problem. The main instrument of the proof is the a priori estimates obtained in the paper.
Keywords: non-classical problem, loaded equation, integral conditions of the second kind, existence and uniqueness of a solution, method of successive approximations, reduction.
Mots-clés : non-local conditions, Goursat problem 
@article{VSGU_2020_26_4_a2,
     author = {A. V. Gilev},
     title = {A nonlocal problem for a hyperbolic equation with a dominant mixed derivative},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {25--35},
     year = {2020},
     volume = {26},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2020_26_4_a2/}
}
TY  - JOUR
AU  - A. V. Gilev
TI  - A nonlocal problem for a hyperbolic equation with a dominant mixed derivative
JO  - Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
PY  - 2020
SP  - 25
EP  - 35
VL  - 26
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VSGU_2020_26_4_a2/
LA  - ru
ID  - VSGU_2020_26_4_a2
ER  - 
%0 Journal Article
%A A. V. Gilev
%T A nonlocal problem for a hyperbolic equation with a dominant mixed derivative
%J Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ
%D 2020
%P 25-35
%V 26
%N 4
%U http://geodesic.mathdoc.fr/item/VSGU_2020_26_4_a2/
%G ru
%F VSGU_2020_26_4_a2
A. V. Gilev. A nonlocal problem for a hyperbolic equation with a dominant mixed derivative. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 26 (2020) no. 4, pp. 25-35. http://geodesic.mathdoc.fr/item/VSGU_2020_26_4_a2/

[1] L. S. Pulkina, Problems with nonclassical conditions for hyperbolic equations, Izdatel-stvo “Samarskii universitet”, Samara, 2012, 194 pp. (In Russ.)

[2] L. S. Pulkina, “The $L_2$ solvability of a nonlocal problem with integral conditions for a hyperbolic equation”, Differential Equations, 36:2 (2000), 316–318 | DOI | Zbl

[3] J. R. Cannon, “The solution of the heat equation subject to the specification of energy”, Quarterly of Applied Mathematics, 21:2 (1963), 155–160 | DOI

[4] L. Byszewski, “Existance and uniqueness of solutions of nonlocal problems for hyperbolic equation $u_{xt} = F(x, t, u, u_x)$”, Journal of Applied Mathematics and Stochastic Analysis, 3:3 (1990), 163–168 https://www.univie.ac.at/EMIS/journals/HOA/JAMSA/Volume3_3/168.pdf | DOI | Zbl

[5] V. A. Ilin, E. I. Moiseev, “Uniqueness of the solution of a mixed problem for the wave equation with nonlocal boundary conditions”, Differential Equations, 36:5 (2000), 728–733 | DOI

[6] D. G. Gordeziani, G. A. Avalishvili, “On the constructing of solutions of the nonlocal initial boundary value problems for one-dimensional medium oscillation equations”, Matem. Mod., 12:1 (2000), 94–103 (In Russ.) | Zbl

[7] A. Bouziani, N. Benouar, “Probleme mixte avec conditions integrales pour une classe d'equations hyperboliques”, Bull. Belg. Math. Soc., 1996, no. 3, 137–145 | DOI | Zbl

[8] A. M. Nakhushev, Loaded equations, their application, Nauka, M., 2012, 232 pp. (In Russ.)

[9] A. I. Kozhanov, L. S. Pulkina, “On the solvability of boundary value problems with a nonlocal boundary condition of integral form for multidimensional hyperbolic equations”, Differential Equations, 42:9 (2006), 1233–1246 | DOI | Zbl

[10] Z. A. Nahusheva, “A nonlocal problem for partial differential equations”, Differential Equations, 22:1 (1986), 171–174 (In Russ.) | Zbl

[11] A. T. Assanova, “Nonlocal problem with integral conditions for a system of hyperbolic equations in characteristic rectangle”, Russian Mathematics, 61:5 (2017), 7–20 | DOI | Zbl

[12] S. G. Mikhlin, Lectures on linear integral equations, Fizmatgiz, M., 1959, 232 pp. (In Russ.)