a spatial decay estimates for a thermoelastic cosserat body without energy dissipation

A cylinder with a prismatic structure is considered and this is &filled& with a thermoelastic material with the cosserat structure. it is supposed that on the lateral surface of the cylinder there are no body forces, or body couple nor heat supply. but, on the base of cylinder, a microrotation is given, which is time-dependent, a displacement, which is also time-dependent, and a thermal displacement is also prescribed. all these loads are assumed to be harmonic functions in time and from their corroborated action, the movement of the body under consideration is induced. we will define a measure associated with the vibration that corresponds to the steady state. assuming that there is a certain critical frequency and we can suppose that any excitation frequency is lower than the critical one, we will be able to obtain an estimation regarding the spatial decay.