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<p>In this paper, we study the existence and stability of solutions to the wave equation with acoustic boundary conditions. The problem is defined in an exterior domain of the disk model of the hyperbolic space. The strategy is change the original problem into a singular one defined in a domain of the unitary ball of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="33_2025_2449_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^N\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. To prove the existence of solution, we use the Faedo–Galerkin method. Moreover, using multipliers techniques we prove the exponential stability of the energy associated with the problem. To overcome the difficulties concerning the singularities, we use a Hardy inequality in a version due to Brezis and Marcus.</p>

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Stability for the wave equation on hyperbolic space with acoustic boundary conditions

  • 期刊论文

  • 发表日期：

    2025年2月17日

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  • Paulo Cesar Carrião,
  • André Vicente

摘要

In this paper, we study the existence and stability of solutions to the wave equation with acoustic boundary conditions. The problem is defined in an exterior domain of the disk model of the hyperbolic space. The strategy is change the original problem into a singular one defined in a domain of the unitary ball of \(\mathbb {R}^N\) R N . To prove the existence of solution, we use the Faedo–Galerkin method. Moreover, using multipliers techniques we prove the exponential stability of the energy associated with the problem. To overcome the difficulties concerning the singularities, we use a Hardy inequality in a version due to Brezis and Marcus.