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<p>We consider a transmission problem for a string composed by three different types of components: a viscoelastic material of memory type, an elastic material (without dissipation) and a Kelvin-Voigt viscoelastic material. The dissipative mechanisms can act on different parts of the string or they can even intersect. The rate of decay will depend on the position of the components in the following sense: if there exists at least one elastic part of the string that connects with the viscoelastic part of Kelvin-Voigt type but not with the memory part, then there is no exponential stability. In fact, the solutions decay polynomially with the rate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(t^{-2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, which is optimal. In every other situation, in which both viscoelastic dampings are present, there exists exponential stability of the solution.</p>

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Asymptotic Stability for Wave Equation with Kelvin-Voigt Damping and Memory Effect

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  • 发表日期：

    2026年6月23日

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  • Antonio Guimarães Leite,
  • María R. Astudillo,
  • Higidio Portillo Oquendo

摘要

We consider a transmission problem for a string composed by three different types of components: a viscoelastic material of memory type, an elastic material (without dissipation) and a Kelvin-Voigt viscoelastic material. The dissipative mechanisms can act on different parts of the string or they can even intersect. The rate of decay will depend on the position of the components in the following sense: if there exists at least one elastic part of the string that connects with the viscoelastic part of Kelvin-Voigt type but not with the memory part, then there is no exponential stability. In fact, the solutions decay polynomially with the rate \(t^{-2}\) t - 2 , which is optimal. In every other situation, in which both viscoelastic dampings are present, there exists exponential stability of the solution.