Semi-uniform stability of operator semigroups and energy decay of damped waves

TL;DR

This paper reviews the concept of semi-uniform stability in $C_0$-semigroups, highlighting its importance and recent developments, and discusses how these results can be applied to determine energy decay rates in damped wave equations.

Contribution

It provides a comprehensive overview of semi-uniform stability theory and its application to energy decay in damped wave problems, emphasizing recent optimal results.

Findings

Overview of semi-uniform stability concepts

Connection between stability and energy decay rates

Application to damped second-order Cauchy problems

Abstract

Only in the last fifteen years or so has the notion of semi-uniform stability, which lies between exponential stability and strong stability, become part of the asymptotic theory of $C_0$-semigroups. It now lies at the very heart of modern semigroup theory. After briefly reviewing the notions of exponential and strong stability, we present an overview of some of the best known (and often optimal) abstract results on semi-uniform stability. We go on to indicate briefly how these results can be applied to obtain (sometimes optimal) rates of energy decay for certain damped second-order Cauchy problems.