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<p>We study a one-dimensional wave equation posed on a two-edge network with constant wave speeds on each edge, coupled at a transmission point and equipped with boundary feedback. A set-valued boundary damping law is imposed at one endpoint and is described by a subset <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Sigma \subset \mathbb {R}^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Σ</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. We aim at determining conditions on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Σ</mi> </math></EquationSource> </InlineEquation> ensuring existence and uniqueness as well as strong and exponential stability of solutions.</p>

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Well-posedness and stability of a transmission problem for the wave equation with set-valued boundary damping

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Fathi Hassine,
  • Olfa Turki

摘要

We study a one-dimensional wave equation posed on a two-edge network with constant wave speeds on each edge, coupled at a transmission point and equipped with boundary feedback. A set-valued boundary damping law is imposed at one endpoint and is described by a subset \(\Sigma \subset \mathbb {R}^{2}\) Σ ⊂ R 2 . We aim at determining conditions on \(\Sigma \) Σ ensuring existence and uniqueness as well as strong and exponential stability of solutions.