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.
Highlights
Let − A be the generator of a C0-semigroup (T (t))t≥0 on a Banach space X
Let − A be the generator of a C0-semigroup (T (t))t≥0 on a Banach space X such that C− ⊆ ρ(A)
Let − A be the generator of a C0-semigroup (T (t))t≥0 on a Banach space X such that C− ⊆ ρ(A), and let Y → X be a continuously embedded Banach space satisfying the following conditions: (1) There exists a CT ≥ 0 such that T (t) ∈ L(Y ) for all t ≥ 0, with T (t) L(Y ) ≤ CT T (t ) L(X); (2) There exists a continuously and densely embedded Banach space Y0 → Y such that [t → e−at T (t) L(Y0,X)] ∈ L1(0, ∞) for all a ∈ (0, ∞)
Summary
Let − A be the generator of a C0-semigroup (T (t))t≥0 on a Banach space X. Versions of Theorem 1.1 for Césaro-type averages have been considered in [32], where numerous counterexamples are presented It was known from [14] that on general Banach spaces (1.3) implies (λ + A)−1 L(X) ≤ C (Re(λ)−α−1 + 1). In Theorem 3.11 and Corollary 3.13 we extend this result and obtain a full characterization of polynomial stability of a semigroup in terms of properties of the resolvent of its generator. We derive versions of Theorem 1.1 on fractional domains, where we make other geometric assumptions on X It is shown in Proposition 3.1 that on a general Banach space X (1.1) implies at most linear growth for semigroup orbits with sufficiently smooth initial values.
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