Geodesic
A~problem with dynamical boundary condition for~a~one-dimensional hyperbolic equation
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 3, pp. 407-423.

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In this paper, we consider a problem with dynamical boundary conditions for a hyperbolic equation. The dynamical boundary condition is a convenient method to take into account the presence of certain damper when fixing the end of a string or a beam. Problems with dynamical boundary conditions containing first-order derivatives with respect to both space and time variables are not self-ajoint, that complicates solution by spectral analysis. However, these difficulties can be overcome by a method proposed in the paper. The main tool to prove the existence of the unique weak solution to the problem is the priori estimates in Sobolev spaces. As a particular example of the wave equation is considered. The exact solution of a problem with dynamical condition is obtained.
Keywords: hyperbolic equation, boundary-value problem, dynamical boundary condition, weak solution
Mots-clés : Sobolev spaces. 
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A. B. Beylin; L. S. Pulkina. A~problem with dynamical boundary condition for~a~one-dimensional hyperbolic equation. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 3, pp. 407-423. http://geodesic.mathdoc.fr/item/VSGTU_2020_24_3_a0/

[1] Tikhonov A. N., Samarskii A. A., Uravneniia matematicheskoi fiziki [Equations of Mathematical Physics], Nauka, Moscow, 2004 (In Russian)

[2] Skubachevskii A. L., Steblov G. M., “On the spectrum of differential operators with a domain that is not dense in $L_2(0,1)$”, Soviet Math. Dokl., 44:3 (1992), 870–875 | MR | Zbl

[3] Ionkin N. I., “The solution of a certain boundary value-problem of the theory of heat conduction with a nonclassical boundary condition”, Differ. Uravn., 13:2 (1977), 294–304 (In Russian) | MR | Zbl

[4] Lažetić N. L., “On the classical solvability of the mixed problem for a second-order one-dimensional hyperbolic equation”, Differ. Equ., 42:8 (2006), 1134–1139 | DOI | MR

[5] Rogozhnikov A. M., “On various kinds of boundary conditions for one-dimensional wave equation”, The Collection of Articles by Young Scientists at the MSU Faculty of Computational Mathematics and Cybernetics, v. 10, 2013, 188–214 (In Russian)

[6] Kirichek V. A., Pulkina L. S., “Problem with dynamic boundary conditions for a hyperbolic equation”, Vestn. Samarsk. Univ. Estestvenno-Nauchn. Ser., 23:1 (2017), 21–27 (In Russian) | Zbl

[7] Korpusov M. O., Razrushenie v neklassicheskikh volnovykh uravneniiakh [Blow-Up in Nonclassical Wave Equations], URSS, Moscow, 2010 (In Russian)

[8] Beylin A. B., Pulkina L. S., “A problem on longitudinal vibrations of a beam with dynamic boundary conditions”, Vestn. Samarsk. Gosud. Univ. Estestvenno-Nauchn. Ser., 2014, no. 3(114), 9–19 (In Russian) | Zbl

[9] Doronin G. G., Lar'kin N. A., Souza A. J., “A hyperbolic problem with nonlinear second-order boundary damping”, Electron. J. Differ. Equ., 1998 (1998), no. 28, 10 pp. https://digital.library.txstate.edu/handle/10877/7927 | Zbl

[10] Andrews K. T., Kuttler K. L., Shillor M., “Second order evolution equations with dynamic boundary conditions”, J. Math. Anal. Appl., 197:3 (1996), 781–795 | DOI | Zbl

[11] Beylin A. B., Pulkina L. S., “A problem on longitudinal vibration in a short bar with dynamical boundary conditions”, Vestn. Samarsk. Univ. Estestvenno-Nauchn. Ser., 23:4 (2017), 7–18 (In Russian) | DOI | Zbl

[12] Louw T., Whitney S., Subramanian A., Viljoen H., “Forced wave motion with internal and boundary damping”, J. Appl. Phys., 111 (2012), 014702 | DOI

[13] Pulkina L. S., “A problem with dynamic nonlocal condition for pseudohyperbolic equation”, Russian Math. (Iz. VUZ), 60:9 (2016), 38–45 | DOI | Zbl

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

[15] Pulkina L. S., Beylin A. B., “Nonlocal approach to problems on longitudinal vibration in a short bar”, Electron. J. Differ. Equ., 2019 (2019), no. 29, 9 pp. https://ejde.math.txstate.edu/Volumes/2019/29/abstr.html | Zbl

[16] Khazanov Kh. S., Mekhanicheskie kolebaniia sistem s raspredelennymi parametrami [Mechanical Vibrations of Systems with Distributed Parameters], Samara State Aerospace Univ., Samara, 2002 (In Russian)

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

[18] Il'in V. A., Tikhomirov V. V., “The wave equation with boundary control at two ends and the problem of the complete damping of a vibration process”, Differ. Equ., 35:5 (1999), 697–708 | MR | Zbl

[19] Il'in V. A., Izbrannye trudy V.A. Il'ina [Selected Works of V.A. Il'in], v. 2, Maks-Press, Moscow, 2008 (In Russian)