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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 2, Pages 276–289 (Mi vsgtu1480)

Differential Equations and Mathematical Physics

A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation

L. S. Pulkinaa, A. E. Savenkovab

a Samara National Research University, Samara, 443086, Russian Federation
b Samara State Technical University, Samara, 443100, Russian Federation

Abstract: 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
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DOI: https://doi.org/10.14498/vsgtu1480

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Bibliographic databases:

UDC: 517.956.3
MSC: 35A01, 35L10, 35A02
Original article submitted 09/III/2016
revision submitted – 22/IV/2016

Citation: L. S. Pulkina, A. E. Savenkova, “A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:2 (2016), 276–289

Citation in format AMSBIB
\Bibitem{PulSav16} \by L.~S.~Pulkina, A.~E.~Savenkova \paper A problem with nonlocal integral condition of the second kind for one-dimensional hyperbolic equation \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2016 \vol 20 \issue 2 \pages 276--289 \mathnet{http://mi.mathnet.ru/vsgtu1480} \crossref{https://doi.org/10.14498/vsgtu1480} \zmath{https://zbmath.org/?q=an:06964487} \elib{https://elibrary.ru/item.asp?id=27126234} 

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This publication is cited in the following articles:
1. A. K. Urinov, Sh. T. Nishonova, “A Problem with Integral Conditions for an Elliptic-Parabolic Equation”, Math. Notes, 102:1 (2017), 68–80
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