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.
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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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The first author is supported by Grant DP160100941 of the Australian Research Council. The second author is supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).
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Rozendaal, J., Veraar, M. Sharp growth rates for semigroups using resolvent bounds. J. Evol. Equ. 18, 1721–1744 (2018). https://doi.org/10.1007/s00028-018-0459-x
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DOI: https://doi.org/10.1007/s00028-018-0459-x