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 is not. When the damping is effective in the whole domain $$(-1,1)$$ , it was proven in Portillo Oquendo and Sanez Pacheco (Appl Math Lett 67:16–20, 2017) that the energy is a non-increasing function of the time variable, with a rate equals 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 the exponential stability, that the semigroup loses speed and it decays polynomially with a slower rate than the one given in Portillo Oquendo and Sanez Pacheco (loc. cit.) [20].
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