Sharp growth rates for semigroups using resolvent bounds

Jan Rozendaal, Mark Veraar*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)
16 Downloads (Pure)


We study growth rates for strongly continuous semigroups. We prove that a growth rate for the resolvent on imaginary lines implies a corresponding growth rate for the semigroup if either the underlying space is a Hilbert space, or the semigroup is asymptotically analytic, or if the semigroup is positive and the underlying space is an (Formula presented.)-space or a space of continuous functions. We also prove variations of the main results on fractional domains; these are valid on more general Banach spaces. In the second part of the article, we apply our main theorem to prove optimality in a classical example by Renardy of a perturbed wave equation which exhibits unusual spectral behavior.

Original languageEnglish
Pages (from-to)1721–1744
Number of pages24
JournalJournal of Evolution Equations
Publication statusPublished - 2018


  • C 0-semigroup
  • Fourier multiplier
  • Kreiss condition
  • Perturbed wave equation
  • Polynomial growth
  • Positive semigroup


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