Decay property for a novel partially dissipative viscoelastic beam system on the real line

We address here a viscoelastic Timoshenko model on the (one-dimensional) real line with memory damping coupled on a shear force. Our main results concern a complete decay structure of the system under the so-called equal wave speeds assumption, as well as without this condition. This is the first result of this type for partially dissipative beam systems with memory-type damping on the shear force. Our method is based on expanded structural conditions such as the so-called SK condition. In addition, we give a characterization of the dissipative structure of the system by using a spectral analysis method, which confirms that our decay structure is optimal.