Abstract:
In this paper, we consider an initial-boundary problem with dynamical nonlocal boundary condition for a pseudohyperbolic fourth-order equation in a rectangular. Dynamical nonlocal boundary condition represents a relation between values of a required solution, its derivatives with respect of spacial variables, second-order derivatives with respect of time-variables and an integral term. This problem may be used as a mathematical model of longitudinal vibration in a thick short bar and illustrates a nonlocal approach to such processes. The main result lies in justification of solvability of this problem. Existence and uniqueness of a generalized solution are proved. The proof is based on the a priori estimates obtained in this paper, Galerkin's procedure and the properties of the Sobolev spaces.
Citation:
A. B. Beylin, L. S. Pulkina, “A problem on longitudinal vibration in a short bar with dynamical boundary conditions”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 2017, no. 4, 7–18
\Bibitem{BeyPul17}
\by A.~B.~Beylin, L.~S.~Pulkina
\paper A problem on longitudinal vibration in a short bar with dynamical boundary conditions
\jour Vestnik SamU. Estestvenno-Nauchnaya Ser.
\yr 2017
\issue 4
\pages 7--18
\mathnet{http://mi.mathnet.ru/vsgu557}
\crossref{https://doi.org/10.18287/2541-7525-2017-23-4-7-18}
\elib{https://elibrary.ru/item.asp?id=32274175}
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https://www.mathnet.ru/eng/vsgu557
https://www.mathnet.ru/eng/vsgu/y2017/i4/p7
This publication is cited in the following 4 articles:
A. V. Gilev, O. M. Kechina, L. S. Pulkina, “Characteristic problem for a fourth-order equation with a dominant derivative”, Vestnik SamU. Estestvenno-Nauchnaya Ser., 27:3 (2021), 14–21
A. B. Beilin, L. S. Pulkina, “Zadacha s dinamicheskim kraevym usloviem dlya odnomernogo giperbolicheskogo uravneniya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 24:3 (2020), 407–423
V. A. Kirichek, “O gladkosti resheniya odnoi nelokalnoi zadachi dlya giperbolicheskogo uravneniya”, Vestn. SamU. Estestvennonauchn. ser., 26:2 (2020), 15–22
A. V. Dyuzheva, “Zadacha s integralnym usloviem I roda dlya uravneniya chetvertogo poryadka”, Vestn. SamU. Estestvennonauchn. ser., 25:1 (2019), 21–31
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