On the Spatial Behavior in Thermoelasticity Without Energy Dissipation

In the present paper we consider a prismatic cylinder occupied by an anisotropic homogeneous compressible linear thermoelastic material that is subject to zero body force and heat supply and zero displacement and thermal displacement on the lateral boundary. The motion is induced by a harmonic time-dependent displacement-thermal displacement specified pointwise over the base. We establish some spatial estimates for some appropriate cross-sectional measures associated with the transient and steady state solutions that describe how they evolves with respect to the axial distance to the excited base. For the transient solutions a domain of influence is observed inside which a measure of the solution decays exponentially with respect to the axial variable, while outside of the domain the solution vanishes. For the harmonic in time vibrations we establish spatial estimates describing how the amplitude of vibration evolves with respect to the axial variable, provided the frequency is lower than a critical value. All results are established for the thermoelastic materials whose constitutive coefficients satisfy strong ellipticity conditions.