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Journal of Evolution Equations

, Volume 18, Issue 4, pp 1721–1744 | Cite as

Sharp growth rates for semigroups using resolvent bounds

  • Jan Rozendaal
  • Mark Veraar
Open Access
Article
  • 67 Downloads

Abstract

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 \(L^{p}\)-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.

Keywords

\(C_{0}\)-semigroup Polynomial growth Positive semigroup Fourier multiplier Kreiss condition Perturbed wave equation 

Mathematics Subject Classification

Primary 47D06 Secondary 34D05 35B40 42B15 

Notes

Acknowledgements

The authors would like to thank Yuri Tomilov for helpful comments, and the anonymous referee for carefully reading the manuscript.

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Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityActonAustralia
  2. 2.Institute of Mathematics, Polish Academy of SciencesWarsawPoland
  3. 3.Delft Institute of Applied MathematicsDelft University of TechnologyDelftThe Netherlands

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