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Abstract <p>This paper considers the asymptotic behavior of solutions of the initial-boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8371_Article_IEq1.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(t\to\infty\)</EquationSource> <!--LobJMat2560830Ishkhanyan-m1--> </InlineEquation>. The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8371_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma(H_{0})\)</EquationSource> <!--LobJMat2560830Ishkhanyan-m2--> </InlineEquation> of the one-dimensional Schrödinger operator <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8371_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(H_{0}\)</EquationSource> <!--LobJMat2560830Ishkhanyan-m3--> </InlineEquation> with periodic coefficients <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8371_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(p(x)\)</EquationSource> <!--LobJMat2560830Ishkhanyan-m4--> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12202_2025_8371_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="33" /> </InlineMediaObject> <EquationSource Format="TEX">\(q(x)\)</EquationSource> <!--LobJMat2560830Ishkhanyan-m5--> </InlineEquation>.</p>

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On the Behavior of Solutions of the Mixed Problem for a Hyperbolic Equation with Periodic Coefficients on the Semi-Axis

  • 期刊论文

  • 发表日期：

    2025年6月1日

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  • A. M. Ishkhanyan,
  • H. A. Matevossian,
  • V. Yu. Smirnov

摘要

Abstract

This paper considers the asymptotic behavior of solutions of the initial-boundary value problem for a second-order hyperbolic equation with periodic coefficients on the semi-axis as \(t\to\infty\) . The main approach to studying the problem under consideration is based on the spectral theory of differential operators, as well as on the properties of the spectrum \(\sigma(H_{0})\) of the one-dimensional Schrödinger operator \(H_{0}\) with periodic coefficients \(p(x)\) and \(q(x)\) .