On the energy decay rates for the 1D damped fractional Klein-Gordon equation

We consider the fractional Klein-Gordon equation in one spatial dimension, subjected to a damping coefficient, which is non-trivial and periodic, or more generally strictly positive on a periodic set. We show that the energy of the solution decays at the polynomial rate Ot-s4-2s for 0<s<2 and at some exponential rate when s >= 2. Our approach is based on the asymptotic theory of C-0 semigroups in which one can relate the decay rate of the energy in terms of the resolvent growth of the semigroup generator. The main technical result is a new observability estimate for the fractional Laplacian, which may be of independent interest.