Non-uniform Stability of Damped Contraction Semigroups

We investigate the stability properties of strongly continuous semigroups generated by operators of the form $A-B{B}^{\ast }$, where $A$ is a generator of a contraction semigroup and $B$ is a possibly unbounded operator. Such systems arise naturally in the study of hyperbolic partial differential equations with damping on the boundary or inside the spatial domain. As our main results we present general sufficient conditions for non-uniform stability of the semigroup generated by $A-B{B}^{\ast }$ in terms of selected observability-type conditions of the pair $(B^{\ast },A)$. We apply the abstract results to obtain rates of energy decay in one-dimensional and two-dimensional wave equations, a damped fractional Klein–Gordon equation and a weakly damped beam equation.