Geodesic
Problem with nonlocal boundary condition for a hyperbolic equation
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3 (2017), pp. 26-33 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper we consider an initial-boundary problem with nonlocal boundary condition for one-dimensional hyperbolic equation. Nonlocal condition is dynamic so as represents a relation between values of derivatives with respect of spacial variables of a required solution, first-order derivatives with respect to time variable and an integral of a required solution of spacial variable. We prove the existence and uniqueness of a generalized solution, which belongs to the Sobolev space. To prove uniquely solvability of the problem techniques developed specifically for research nonlocal problems are used. The application of these methods allowed us to obtain a priori estimates, through which the uniqueness of the solution is proved. The proof is based on the a priori estimates obtained in this paper and Galyorkin's procedure.
Keywords: nonlocal boundary condition, hyperbolic equation, generalized solution, Sobolev space. 
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     author = {V. A. Kirichek},
     title = {Problem with nonlocal boundary condition for a hyperbolic equation},
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V. A. Kirichek. Problem with nonlocal boundary condition for a hyperbolic equation. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 3 (2017), pp. 26-33. http://geodesic.mathdoc.fr/item/VSGU_2017_3_a3/

[1] Tikhonov A. N., Samarsky A. A., Equations of mathematical physics, Nauka, M., 1977, 728 pp. (in Russian)

[2] T. Louw, S. Whitney, A. Subramanian, H. Viljoen, “Forced wave mation with internal and boundary damping”, Journal of applied physics, 111 (2012), 014702 (in English) | DOI

[3] Korpusov M. O., Destruction in nonclassical wave equations, URSS, M., 2010, 240 pp. (in Russian)

[4] Doronin G. G., Lar'kin N. A., Souza A. J., “A hyperbolic problem with nonlinear second-order boundary damping”, EJDE, 1998, no. 28, 1–10 (in English) | MR

[5] Beylin A. B., Pulkina L. S., “Task on longitudinal vibrations of a rod with dynamic boundary conditions”, Vestnik of Samara State University, 2014, no. 3(114), 9–19 (in Russian)

[6] Pulkina L. S., “A problem with dynamic nonlocal condition for pseudohyperbolic equation”, Russian Mathematics (Iz. VUZ), 2016, no. 9, 42–50 (in Russian)

[7] Ladyzhenskaya O. A., Boundary problems of mathematical physics, Nauka, M., 1973, 407 pp. (in Russian)

[8] Beylin S. A., “On a certain problem for a wave equation”, Vestnik of Samara State University, 2011, no. 5(86), 12–17 (in Russian)

[9] Rogozhnikov A. M., “On various types of boundary conditions for the one-dimensional equation of oscillations”, Collection of articles of young scientists of the faculty of Computational Mathematics and Cybernetics of MSU, 10, 2013, 188–214 (in Russian)

[10] Cannon J. R., “The solution of tne heat equation subject to the specification of energy”, Quart. Appl. Math., 21 (1963), 155–160 (in English) | DOI | MR

[11] Kamynin L. I., “A boundary value problem in the theory of heat conduction with a nonclassical boundary condition”, Computational Mathematics and Mathematical Physics, 4:6 (1964), 1006–1024 (in Russian)

[12] Pulkina L. S., “A nonclassical problem for a degenerate hyperbolic equation”, Russian Mathematics (Iz. VUZ), 1991, no. 11, 48–51 (in Russian)

[13] Pulkina L. S., “Certain nonlocal problem for a degenerate hyperbolic equation”, Mathematical notes, 51:3 (1992), 91–96 (in Russian)