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arXiv · 1712.00692

Sharp growth rates for semigroups using resolvent bounds

Jan Rozendaal·Mark Veraar
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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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Jan Rozendaal, Mark Veraar. 2017-12-03. Sharp growth rates for semigroups using resolvent bounds. https://doi.org/10.1007/s00028-018-0459-x

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