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A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation. (Russian. English summary) Zbl 1413.35007
Summary: 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.

MSC:
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35L10 Second-order hyperbolic equations
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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[1] [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) · Zbl 1027.74505
[2] [2] Bouziani A., “On the solvability of a nonlocal problems arising in dynamics of moisture transfer”, Georgian Mathematical Journal, 10:4 (2003), 607–622 · Zbl 1059.35072
[3] [3] Pulkina L. S., “A nonlocal problem with integral conditions for a hyperbolic equation”, Differ. Equ., 40:7 (2004), 947–953 · Zbl 1077.35076
[4] [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 · Zbl 1141.35408
[5] [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 1135.35002
[6] [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)
[7] [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] [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 · Zbl 1227.35015
[9] [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 · Zbl 1255.35158
[10] [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] [11] Pulkina L. S., “Solution to nonlocal problems of pseudohyperbolic equations”, EJDE, 2014:116 (2014), 1–9 · Zbl 1297.35147
[12] [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) · Zbl 0189.40603
[13] [13] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki [Boundary Value Problems of Mathematical Physics], Nauka, Moscow, 1973, 402 pp. (In Russian) · Zbl 0284.35001
[14] [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. · Zbl 0111.29008
[15] [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 · Zbl 1423.74389
[16] [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
[17] [17] Korpusov M. O., Razrushenie v neklassicheskikh volnovykh uravneniiakh [Blow-up in nonclassical wave equations], URSS, Moscow, 2010, 237 pp. (In Russian)
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