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Zh. Mat. Fiz. Anal. Geom., 2018, Volume 14, Number 4, Pages 393–405 (Mi jmag705)  

Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time

Anatoly Anikinab, Sergey Dobrokhotovab, Vladimir Nazaikinskiiab

a Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Moscow Region, 141701, Russia
b Ishlinsky Institute for Problems in Mechanics RAS, pr. Vernadskogo, 101-1, Moscow, 119526, Russia

Abstract: We study the Cauchy problem for the inhomogeneous two-dimensional wave equation with variable coefficients and zero initial data. The right-hand side is assumed to be localized in space and time. The equation is considered in a domain with a boundary (shore). The velocity is assumed to vanish on the shore as a square root of the distance to the shore, that is, the wave equation has a singularity on the curve. This curve determines the boundary of the domain where the problem is studied. The main result of the paper is efficient asymptotic formulas for the solution of this problem, including the neighborhood of the shore.

Key words and phrases: wave equation, asymptotic solution, Maslov's canonical operator.

Funding Agency Grant Number
Russian Science Foundation 16-11-10282
This work is supported by Russian Science Foundation, project No. 16-11-10282.


DOI: https://doi.org/10.15407/mag14.04.393

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Document Type: Article
MSC: 34E20, 35L05, 35Q35
Received: 19.03.2018
Language: English

Citation: Anatoly Anikin, Sergey Dobrokhotov, Vladimir Nazaikinskii, “Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time”, Zh. Mat. Fiz. Anal. Geom., 14:4 (2018), 393–405

Citation in format AMSBIB
\Bibitem{AniDobNaz18}
\by Anatoly~Anikin, Sergey~Dobrokhotov, Vladimir~Nazaikinskii
\paper Asymptotic solutions of the wave equation with degenerate velocity and with right-hand side localized in space and time
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2018
\vol 14
\issue 4
\pages 393--405
\mathnet{http://mi.mathnet.ru/jmag705}
\crossref{https://doi.org/10.15407/mag14.04.393}


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