Geodesic
A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 2, pp. 276-289.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider a problem for a one-dimensional hyperbolic equation with nonlocal integral condition of the second kind. Uniqueness and existence of a generalized solution are proved. In order to prove this statement we suggest a new approach. The main idea of it is that given nonlocal integral condition is equivalent with a different condition, nonlocal as well but this new condition enables us to derive a priori estimates of a required solution in Sobolev space. By means of derived estimates we show that a sequence of approximate solutions constructed by Galerkin procedure is bounded in Sobolev space. This fact implies the existence of weakly convergent subsequence. Finally, we show that the limit of extracted subsequence is the required solution to the problem.
Keywords: hyperbolic equation, nonlocal integral conditions, generalized solution, Sobolev space, Galerkin procedure. 
@article{VSGTU_2016_20_2_a5,
     author = {L. S. Pulkina and A. E. Savenkova},
     title = {A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {276--289},
     publisher = {mathdoc},
     volume = {20},
     number = {2},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2016_20_2_a5/}
}
TY  - JOUR
AU  - L. S. Pulkina
AU  - A. E. Savenkova
TI  - A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2016
SP  - 276
EP  - 289
VL  - 20
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2016_20_2_a5/
LA  - ru
ID  - VSGTU_2016_20_2_a5
ER  - 
%0 Journal Article
%A L. S. Pulkina
%A A. E. Savenkova
%T A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2016
%P 276-289
%V 20
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2016_20_2_a5/
%G ru
%F VSGTU_2016_20_2_a5
L. S. Pulkina; A. E. Savenkova. A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 20 (2016) no. 2, pp. 276-289. http://geodesic.mathdoc.fr/item/VSGTU_2016_20_2_a5/

[1] Gordeziani D. G., Avalishvili G. A., “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 Russian) | MR | Zbl

[2] Bouziani A., “On the solvability of a nonlocal problems arising in dynamics of moisture transfer”, Georgian Mathematical Journal, 10:4 (2003), 607–622 | DOI | MR | Zbl

[3] Pulkina L. S., “A nonlocal problem with integral conditions for a hyperbolic equation”, Differ. Equ., 40:7 (2004), 947–953 | DOI | MR | Zbl

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

[5] Nakhushev A. M., Zadachi so smeshcheniem dlia uravnenii v chastnykh proizvodnykh [Problems with shifts for partial differential equations], Nauka, Moscow, 2006, 288 pp. (In Russian) | Zbl

[6] Dmitriev V. B., “A nonlocal problem with integral conditions for the wave equation”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2006, no. 2(42), 15–27 (In Russian) | MR

[7] Strigun M. V., “On certain nonlocal problem with integral boundary condition for hyperbolic equation”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2009, no. 8(74), 78–87 (In Russian)

[8] Avalishvili G., Avalishvili M., Gordeziani D., “On integral nonlocal boundary problems for some partial differential equations”, Bull. Georg. Natl. Acad. Sci., 5:1 (2011), 31–37 | MR | Zbl

[9] Pulkina L. S., “Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind”, Russian Math. (Iz. VUZ), 56:4 (2012), 62–69 | DOI | MR | Zbl

[10] Pulkina L. S., Zadachi s neklassicheskimi usloviiami dlia giperbolicheskikh uravnenii [Problems with non-classical conditions for hyperbolic equations], Samara University, Samara, 2012, 194 pp. (In Russian)

[11] Pulkina L. S., “Solution to nonlocal problems of pseudohyperbolic equations”, EJDE, 2014:116 (2014), 1–9 http://ejde.math.txstate.edu/Volumes/2014/116/pulkina.pdf | MR

[12] Lions J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires [Some Methods for Solving Nonlinear Boundary Value Problems], Etudes mathematiques, Dunod, Paris, 1969, xx+554 pp. (In French) | MR | Zbl

[13] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki [Boundary Value Problems of Mathematical Physics], Nauka, Moscow, 1973, 402 pp. (In Russian) | MR | Zbl

[14] Tikhonov A. N.; Samarskii A. A., Equations of mathematical physics, International Series of Monographs on Pure and Applied Mathematics, 39, Pergamon Press., Oxford etc., 1963, xvi+765 pp. | MR | Zbl

[15] Fedotov I. A., Polyanin A. D., Shatalov M. Yu., “Theory of free and forced vibrations of a rigid rod based on the Rayleigh model”, Dokl. Phys., 52:11 (2007), 607–612 | DOI | MR | Zbl

[16] Doronin G. G., Lar'kin N. A., Souza A. J., “A hyperbolic problem with nonlinear second-order boundary damping”, EJDE, 1998:28 (1998), 1–10 http://ejde.math.txstate.edu/Volumes/1998/28/Doronin.pdf

[17] Korpusov M. O., Razrushenie v neklassicheskikh volnovykh uravneniiakh [Blow-up in nonclassical wave equations], URSS, Moscow, 2010, 237 pp. (In Russian)