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Journal of Evolution Equations

, Volume 18, Issue 4, pp 1633–1674 | Cite as

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

  • Reinhard Stahn
Article
  • 36 Downloads

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 \(\hat{f}\). 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 \(\hat{f}\) having a singularity of logarithmic type at zero, one novelty in our paper is that we assume \(\hat{f}\) 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.

Keywords

\(C_0\)-semigroup Wave equation (local) rate of decay Laplace transform Logarithmic singularity Tauberian theorem 

Mathematics Subject Classification

40E05 44A10 (47D06, 35B40) 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für Analysis, TU DresdenDresdenGermany

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