Optimal rates of decay for operator semigroups on Hilbert spaces

TL;DR

This paper establishes optimal decay rates for $C_0$-semigroups on Hilbert spaces based on resolvent growth, providing both necessary and sufficient conditions, and applies these results to wave equations with boundary damping.

Contribution

It extends existing results by characterizing when optimal decay rates are achievable and offers new asymptotic estimates, especially for normal operators and boundary damping scenarios.

Findings

Optimal decay rates are characterized by resolvent growth conditions.

The conditions are both necessary and sufficient for a large class of semigroups.

New sharper decay estimates are provided for wave equations with boundary damping.

Abstract

We investigate rates of decay for $C_0$-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. 853-929, 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.