Local decay of $C_0$-semigroups with a possible singularity of logarithmic type at zero

Abstract

We prove decay rates for a vector-valued function $f$ of a non-negative real variable with bounded weak derivative, under rather general conditions on the Laplace transform $\hat{f}$. This generalizes results of Batty-Duyckaerts (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 non-decreasing function $M$ and satisfying a growth assumption with respect to a different non-decreasing 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.

37 pages. This is a significantly enhanced version of my paper "A quantified Tauberian theorem and local decay of C0-semigroups" submitted to arxiv in May 2017

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• Optimal rates of decay for operator semigroups on Hilbert spaces
• Quantified versions of Ingham's theorem
• A quantified Tauberian theorem for sequences
• A quantified Tauberian theorem and local decay of $C_0$-semigroups
• A quantified Tauberian theorem for Laplace-Stieltjes transform

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• Optimality of the quantified Ingham-Karamata theorem for operator semigroups with general resolvent growth
• On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: a constructive approach
• An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions
• Resolvent estimates for the one-dimensional damped wave equation with unbounded damping
• Optimal decay of semi-uniformly stable operator semigroups with empty spectrum
• On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem