Existence and stability of time periodic solutions to nonlinear elastic wave equations with viscoelastic terms

This paper studies a nonlinear viscoelastic equation with a time periodic external force on the three dimensional whole space. The existence of a time periodic solution is proved by using a spectral decomposition and the Poincaré map when the external force is small enough. Based on the regularity estimates of the time periodic solution derived from the smoothing effect of the semigroup, a stability result is obtained with time decay estimates of perturbations.