Sharp growth rates for semigroups using resolvent bounds

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.

Locations

• arXiv (Cornell University) - View - PDF
• Data Archiving and Networked Services (DANS) - View - PDF
• ANU Open Research (Australian National University) - View - PDF
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• Journal of Evolution Equations - View - PDF