We investigate rates of decay for C0-semigroups on Hilbert spaces under assumptions on the resolvent growth of the semigroup generator. Our main results show that one obtains the best possible estimate on the rate of decay, that is to say an upper bound which is also known to be a lower bound, under a comparatively mild assumption on the growth behaviour. This extends several statements obtained by Batty, Chill and Tomilov (J. Eur. Math. Soc., vol. 18(4), pp. 2016). In fact, for a large class of semigroups our condition is not only sufficient but also necessary for this optimal estimate to hold. Even without this assumption we obtain a new quantified asymptotic result which in many cases of interest gives a sharper estimate for the rate of decay than was previously available, and for semigroups of normal operators we are able to describe the asymptotic behaviour exactly. We illustrate the strength of our theoretical results by using them to obtain sharp estimates on the rate of energy decay for a wave equation subject to viscoelastic damping at the boundary.2010 Mathematics Subject Classification. 47D06, 34D05, 34G10 (35B40, 35L05, 26A12).

We consider the damped wave equation with Dirichlet boundary conditions on the unit square parametrized by Cartesian coordinates x and y. We assume the damping a to be strictly positive and constant for x < σ and zero for x > σ. We prove the exact t −4/3 -decay rate for the energy of classical solutions. Our main result (Theorem 1) answers question (1) of [1, Section 2C.]. MSC2010: Primary 35B40, 47D06. Secondary 35L05, 35P20.

We consider the wave equation with a boundary condition of memory type. Under natural conditions on the acoustic impedancek of the boundary one can define a corresponding semigroup of contractions [9]. With the help of Tauberian theorems we establish energy decay rates via resolvent estimates on the generator −A of the semigroup. We reduce the problem of estimating the resolvent of −A to the problem of estimating the resolvent of the corresponding stationary problem. Under not too strict additional assumptions on k we establish an upper bound on the resolvent. For the wave equation on the interval or the disk or for certain acoustic impedances making 0 a spectral point of A we prove our estimates to be sharp.Let e τ (t) = e −τ t 1 [0,∞) (t) andMSC2010: Primary 35B40, 35L05. Secondary 35P20, 47D06. We use the convention to identify functions defined on the interval [0, ∞) with functions defined on R but zero to the left of t = 0. 1 2 Poincaré inequality: If Ω is a bounded Lipschitz domain then there exists a C > 0 such that for all p ∈ H 1 (Ω) with Ω p = 0 we have Ω |p| 2 ≤ C Ω |∇p| 2 .3 Here and in the following we abbreviate L p ν ((0, ∞)τ ; L 2 (∂Ω)) simply by L p ν for p ∈ {1, 2}.

We investigate the stability properties of strongly continuous semigroups generated by operators of the form A − BB * , 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 − BB * in terms of selected observability-type conditions of the pair (B * , A). We apply the abstract results to obtain rates of energy decay in onedimensional and two-dimensional wave equations, a damped fractional Klein-Gordon equation and a weakly damped beam equation.

No abstract