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Decay of $$C_0$$ C 0 -semigroups and local decay of waves on even (and odd) dimensional exterior domains

Decay of $$C_0$$ C 0 -semigroups and local decay of waves on even (and odd) dimensional exterior... We prove decay rates for a vector-valued function f of a nonnegative real variable with bounded weak derivative, under rather general conditions on the Laplace transform $$\hat{f}$$ f ^ . This generalizes results of Batty and Duyckaerts (J Evol Equ 8(4):765–780, 2008) and other authors in later publications. Besides the possibility of $$\hat{f}$$ f ^ having a singularity of logarithmic type at zero, one novelty in our paper is that we assume $$\hat{f}$$ f ^ to extend to a domain to the left of the imaginary axis, depending on a nondecreasing function M and satisfying a growth assumption with respect to a different nondecreasing function K. The decay rate is expressed in terms of M and K. We prove that the obtained decay rates are essentially optimal for a very large class of functions M and K. Finally, we explain in detail how our main result improves known decay rates for the local energy of waves on exterior domains. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Decay of $$C_0$$ C 0 -semigroups and local decay of waves on even (and odd) dimensional exterior domains

Stahn, Reinhard
Journal of Evolution Equations , Volume 18 (4) – May 28, 2018
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References (32)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-018-0455-1
Publisher site
See Article on Publisher Site

Abstract

We prove decay rates for a vector-valued function f of a nonnegative real variable with bounded weak derivative, under rather general conditions on the Laplace transform $$\hat{f}$$ f ^ . This generalizes results of Batty and Duyckaerts (J Evol Equ 8(4):765–780, 2008) and other authors in later publications. Besides the possibility of $$\hat{f}$$ f ^ having a singularity of logarithmic type at zero, one novelty in our paper is that we assume $$\hat{f}$$ f ^ to extend to a domain to the left of the imaginary axis, depending on a nondecreasing function M and satisfying a growth assumption with respect to a different nondecreasing function K. The decay rate is expressed in terms of M and K. We prove that the obtained decay rates are essentially optimal for a very large class of functions M and K. Finally, we explain in detail how our main result improves known decay rates for the local energy of waves on exterior domains.

Journal

Journal of Evolution Equations – Springer Journals

Published: May 28, 2018

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