Abstract

Only in the last 15 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 C0-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. This article is part of the theme issue 'Semigroup applications everywhere'.

Highlights

  • Exponential and strong stability of C0-semigroups are two classical topics in semigroup theory, and the literature on these topics, through various deep results over the past fifty years, has reached a reasonably complete state; we refer to [10,11,114] for extensive accounts

  • 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 C0-semigroups

  • After briefly reviewing the notions of exponential and strong stability, we present an overview of some of the best known abstract results on semi-uniform stability

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Summary

Introduction

Exponential and strong stability of C0-semigroups are two classical topics in semigroup theory, and the literature on these topics, through various deep results over the past fifty years, has reached a reasonably complete state; we refer to [10,11,114] for extensive accounts. Concerning the closely related situation of so-called cut-off (or Lax-Phillips) semigroups it is important to note that around the same time as Batty and Duyckaerts in [22], Christianson proved a very similar decay rate result for functions of the form f (t) = χ1T (t)R(1, A)kχ, where (T (t))t≥0 is a C0-semigroup of contractions on a Hilbert space, χ1 and χ2 are bounded linear operators, the Laplace transform f extends to a domain ΩM where it satisfies a polynomial growth estimate with K(s) = C(1 + s)N , and k ≥ N + 2 [46, Theorem 3]. The paper contains versions of Theorem 3.5 for unbounded semigroups; see e. g. [132, Section 4]

Applications to second-order Cauchy problems
Concluding remarks

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