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In this paper, we consider two systems of type Rao-Nakra sandwich beam in thewhole line R with a frictional damping or an infinite memory acting on theEuler-Bernoulli equation. When the speeds of propagation of the two waveequations are equal, we show that the solutions do not converge to zero whentime goes to infinity. In the reverse situation, we prove some L2(R)-norm andL1(R)-norm decay estimates of solutions and theirs higher order derivativeswith respect to the space variable. Thanks to interpolation inequalities andCarlson inequality, these L2(R)-norm and L1(R)-norm decay estimates lead tosimilar ones in the Lq(R)-norm, for any q>1. In our both L2(R)-norm andL1(R)-norm decay estimates, we specify the decay rates in terms of theregularity of the initial data and the nature of the control.

Some Lq(R)-norm decay estimates for two Cauchy systems of type Rao-Nakra sandwich beam with a frictional damping or an infinite memory