Exponential decay of energy of the Euler-Bernoulli beam with locally distributed Kelvin-Voigt damping

In this paper, we consider the longitudinal and transversal vibrations of the Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam. We prove that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is not exponentially stable. Due to the locally distributed and unbounded nature of the damping, we use a frequency domain method and combine a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent. We also show that the associated semigroups are not analytic.