Wave Equation With Kelvin–Voigt and Frictional Damping: Analysis of Asymptotic Stability

We analyze a transmission problem for a string composed of three distinct types of materials: an elastic material with frictional damping, an elastic (non-dissipative) material, and a Kelvin–Voigt viscoelastic material. The dissipative mechanisms may act on separate parts of the string or overlap in certain regions. The decay rate of the solutions depends on the arrangement of these components. Specifically, if there exists an elastic segment that connects with the Kelvin–Voigt viscoelastic material without being in contact with the frictional damping region, exponential stability is not achieved. Instead, the solutions exhibit polynomial decay at an optimal rate of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:math> . In all other configurations, where both damping mechanisms are present, the solutions are exponentially stable.