Optimal rates of decay for operator semigroups on Hilbert spaces

Abstract

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).

Refined scales of decaying rates of operator semigroups on Hilbert spaces: Typical behavior

“…As discussed previously, after Batty and Duyckaerts [4] have related the decaying rates of stable bounded C 0 -semigroups to the arbitrary growth of the norms of the respective resolvents, the study of polynomial and logarithmic scales of such rates has been the subject of many recent papers (see [3,6,16] and references therein). In contrast with this setting, our first result says that the decaying rates of the orbits of normal C 0 -semigroups of contractions, typically in Baire's sense, may depend on sequences of time going to infinity.…”

“…Telephone +55 16 3351 9153, fax +55 16 3361 2081. applications of the theory to PDEs; namely, estimates on the norm of the resolvent of the generator are often easier to compute than the estimates on the norm of the semigroup itself. In this context, we refer to [1,3,7,8,16], among others. An important intermediate step from Gearhart-Prüss Theorem to results by Rozendaal et al [16] was the Batty-Duyckaerts Theorem (Theorem 1.1 below) which relates the decaying rates of T (t)A −1 B(X) , iR ⊂ ̺(A), with the arbitrary growth of the norm of the resolvent of the generator.…”

Second Order Linear Evolution Equations with General Dissipation

“…III. An intriguing and possibly challenging task would be to investigate semiuniform (or semiuniform-like) decay rates of S(t) which are not necessarily of polynomial type, making use of recent abstract results obtained in [5] (see also [39]) dealing with fine decay scales of strongly continuous semigroups.…”

Optimality of the quantified Ingham–Karamata theorem for operator semigroups with general resolvent growth

“…On the other hand, it was shown in [5] that if M (s) = K(s) = s α , s ≥ 1, for some α > 0 and if X is a Hilbert space then (1.4) may be replaced by the optimal estimate T (t)A −1 = O(t −1/α ), t → ∞. This Hilbert space result has subsequently been extended, first in [3] and then rather substantially in [13]. On the other hand, it was also shown in [5] that in the above polynomial case the upper bound in (1.4) is sharp if we impose no restrictions on the Banach space X; see also [2,14].…”