Some [formula omitted]-norm and [formula omitted]-norm decay estimates for Cauchy Timoshenko type systems with a frictional damping or an infinite memory

Abstract

In this paper, we consider two Timoshenko-type systems in the whole line R with a frictional damping or an infinite memory acting on the equation that describes the shear angle displacements. We prove some L2(R)-norm and L1(R)-norm decay estimates of solutions and their higher-order derivatives with respect to the space variable. Thanks to interpolation inequalities, these L2(R)-norm and L1(R)-norm decay estimates lead to similar ones in the Lp(R)-norm, for any p∈]1,2[∪]2,+∞].