Exponential decay of energy of vibrating strings with local viscoelasticity

In our previous paper, we studied the energy decay rate for an one-dimensional linear wave equation with the Kelvin-Voigt damping presented on a subinterval which models an elastic string with one segment made of viscoelastic material and the other of elastic material. It was proved that the energy of that system does not decay exponentially when each segment is homogeneous, i.e., coefficient functions are piecewise constant and have discontinuity at the interface. This is a puzzling result since it is not seen for the local viscous damping, where the well-known "geometric optics" condition applies. In order to get a better understanding of the causes for this phenomenon, we study two related problems in this paper. We first reconsider the above system with smooth coefficient functions. Then we replace the Kelvin-Voigt model by the Boltzmann model and allow discontinuity of material properties at the interfaces. Exponential energy decay is proved for both cases. These new results suggest that discontinuity of material properties at the interface and the "type" of the damping can affect the qualitative behavior of the energy decay.