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

TL;DR

This paper investigates decay estimates for solutions of Rao-Nakra sandwich beam systems with damping or memory effects, revealing conditions for decay and providing explicit decay rates in various norms based on initial data regularity.

Contribution

It establishes Lq(R)-norm decay estimates for Rao-Nakra systems with damping or memory, extending previous results by including higher order derivatives and different norm estimates.

Findings

Solutions do not decay when wave speeds are equal.

Decay estimates are obtained for solutions and derivatives in L2 and L1 norms.

Decay rates depend on initial data regularity and control nature.

Abstract

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