Gevrey semigroup of the type III localized thermoelastic model

Abstract

We consider the Euler Bernoulli beam model with localized thermoelastic component following the Green and Naghdi type III theory, see [6–9]. We prove that the corresponding semigroup is of Gevrey class 8, regardless of the size of the thermoelastic component or the position it occupies over the beam. In particular, our result implies the instantaneous smoothing effect property over the initial data, the exponential stability of the semigroup and that the exponential rate is equal to the spectral bound of its infinitesimal generator.