Spectrum and Stability for Elastic Systems with Global or Local Kelvin--Voigt Damping

In this paper, we study the mathematical properties of a variational second order evolution equation, which includes the equations modelling vibrations of the Euler--Bernoulli and Rayleigh beams with the global or local Kelvin--Voigt (K--V) damping. In particular, our results describe the semigroup setting, the strong asymptotic stability and exponential stability of the semigroup, the analyticity of the semigroup, as well as characteristics of the spectrum of the semigroup generator under various conditions on the damping. We also give an example to show that the energy of a vibrating string does not decay exponentially when the K--V damping is distributed only on a subinterval which has one end coincident with one end of the string.