Stability for coupled waves with locally disturbed Kelvin-Voigt damping

TL;DR

This paper investigates the energy decay rates in a coupled wave system with partial Kelvin-Voigt damping, revealing slower polynomial decay due to coupling effects and non-smooth damping coefficients.

Contribution

It demonstrates how coupling and non-smooth damping coefficients affect the polynomial decay rate of energy in the system.

Findings

Energy decays polynomially at a rate at least as slow as t^(-1/12)

Lack of exponential stability in the system

Slower decay rate compared to full damping scenarios

Abstract

We consider a coupled wave system with partial Kelvin-Voigt damping in the interval (-1,1), where one wave is dissipative and the other does not. When the damping is effective in the whole domain (-1,1) it was proven in H.Portillo Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt damping, Applied Mathematics Letters 67 (2017), 16-20. That the energy is decreasing over the time with a rate equal to $t^{-\frac{1}{2}}$. In this paper, using the frequency domain method we show the effect of the coupling and the non smoothness of the damping coefficient on the energy decay. Actually, as expected we show the lack of exponential stability, that the semigroup loses speed and it decays polynomially with a slower rate then given in, H.Portillo Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt damping, Applied Mathematics Letters 67 (2017), 16-20, down to zero at least as $t^{-\frac{1}{12}}$.