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We consider an abstract system of two coupled evolution equations. One of these equations has an internal damping, and the other is simply elastic. When both equations have the same propagation speed, Alabau et al. (J Evol Equ 2:127–150, 2002) showed that the semigroup of this system decays polynomially in time with the rate $$t^{-1/2}$$ t - 1 / 2 . In this work, we consider this coupled...