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.

20 pages. To appear in Journal of Evolution Equations

References in corpus (2)

• Fourier multiplier theorems involving type and cotype
• Fourier multiplier theorems on Besov spaces under type and cotype conditions

Cited by in corpus (6)

• Semi-uniform stability of operator semigroups and energy decay of damped waves
• Operator-valued $(L^p,L^q)$ Fourier multipliers and stability theory for evolution equations
• Stability of uniformly eventually positive $C_0$-semigroups on $L_p$-spaces
• Functional calculus on weighted Sobolev spaces for the Laplacian on the half-space
• Improved polynomial decay for unbounded semigroups
• Resolvent estimates for the one-dimensional damped wave equation with unbounded damping