Best rates of decay for coupled waves with different propagation speeds

Abstract

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 $$t^{-1/2}$$ . In this work, we consider this coupled system when the propagation speeds of the equations are different, and we study the asymptotic behavior of the semigroup. For this case, we show that the semigroup still decays polynomially with a slower rate as $$t^{-1/4}$$ . Moreover, we prove that this rate of decay is the best.