J. Evol. Equ. © 2018 Springer International Publishing AG, Journal of Evolution part of Springer Nature Equations https://doi.org/10.1007/s00028-018-0455-1 Decay of C -semigroups and local decay of waves on even (and odd) dimensional exterior domains Reinhard Stahn 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 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 f having a singularity of logarithmic type at zero, one novelty in our paper is that we assume 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. 1. Introduction In the last decade,
Journal of Evolution Equations – Springer Journals
Published: May 28, 2018
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