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.
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Wolfgang Arendt, Charles Batty, Matthias Hieber, and Frank Neubrander. Vector-valued Laplace transforms and Cauchy problems. Birkhäuser, Basel, 2nd edition, 2011.
András Bátkai, Klaus-Jochen Engel, Jan Prüss, and Roland Schnaubelt. Polynomial stability of operator semigroups. Math. Nachr., 279(13-14):1425–1440, 2006.
Charles Batty, Alexander Borichev, and Yuri Tomilov. \(L^{p}\)-Tauberian theorems and \(L^{p}\)-rates for energy decay. J. Funct. Anal., 270(3):1153–1201, 2016.
Charles Batty and Thomas Duyckaerts. Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ., 8(4):765–780, 2008.
Charles J.K. Batty, Ralph Chill, and Yuri Tomilov. Fine scales of decay of operator semigroups. J. Eur. Math. Soc. (JEMS), 18(4):853–929, 2016.
Jean-François Bony and Vesselin Petkov. Resolvent estimates and local energy decay for hyperbolic equations. Ann. Univ. Ferrara, Sez. VII, Sci. Mat., 52(2):233–246, 2006.
Alexander Borichev and Yuri Tomilov. Optimal polynomial decay of functions and operator semigroups. Math. Ann., 347(2):455–478, 2010.
Nicolas Burq. Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math., 180(1):1–29, 1998.
Ralph Chill. Tauberian theorems for vector-valued Fourier and Laplace transforms. Stud. Math., 128(1):55–69, 1998.
Ralph Chill and David Seifert. Quantified versions of Ingham’s theorem. Bull. Lond. Math. Soc., 48(3):519–532, 2016.
Ralph Chill and Yuri Tomilov. Stability of operator semigroups: ideas and results. In Perspectives in operator theory. Papers of the workshop on operator theory, Warsaw, Poland, April 19–May 3, 2004, pages 71–109. Warsaw: Polish Academy of Sciences, Institute of Mathematics, 2007.
Gregory Debruyne and Jason Vindas. Note on the absence of remainders in the Wiener-Ikehara theorem. arXiv:1801.03491, 2018.
Markus Hartlapp. A quantified Tauberian theorem for the Laplace-Stieltjes transform. Arch. Math. 2018. https://doi.org/10.1007/s00013-018-1164-2.
Mitsuru Ikawa. Decay of solutions of the wave equation in the exterior of several convex bodies. Ann. Inst. Fourier, 38(2nd):113–146, 1988.
Jacob Korevaar. Tauberian theory. A century of developments. Springer, Berlin, 2004.
Peter D. Lax and Ralph S. Phillips. Scattering theory. Pure and Applied Mathematics (New York) 26. New York, London: Academic Press, xii, 276 p., 1967.
Zhuangyi Liu and Bopeng Rao. Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys., 56(4):630–644, 2005.
María M. Martínez. Decay estimates of functions through singular extensions of vector-valued Laplace transforms. J. Math. Anal. Appl., 375(1):196–206, 2011.
Donald J. Newman. Simple analytic proof of the prime number theorem. Am. Math. Mon., 87:693–696, 1980.
Georgi Popov and Georgi Vodev. Distribution of the resonances and local energy decay in the transmission problem. Asymptotic Anal., 19(3-4):253–265, 1999.
Jan Rozendaal, David Seifert, and Reinhard Stahn. Optimal rates of decay for operator semigroups on Hilbert spaces. arXiv:1709.08895, submitted, 2017.
Jan Rozendaal and Mark Veraar. Stability theory for semigroups using \((L^p,L^q)\) Fourier multipliers. arXiv:1710.00891, submitted, 2017.
David Seifert. A quantified Tauberian theorem for sequences. Studia Math., 227(2):183–192, 2015.
David Seifert. Rates of decay in the classical Katznelson-Tzafriri theorem. J. Anal. Math., 130:329–354, 2016.
Reinhard Stahn. A quantified Tauberian theorem and local decay of \(C_0\)-semigroups. arXiv:1705.03641, 2017.
Georgi Vodev. On the uniform decay of the local energy. Serdica Math. J., 25(3):191–206, 1999.
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Stahn, R. Decay of \(C_0\)-semigroups and local decay of waves on even (and odd) dimensional exterior domains. J. Evol. Equ. 18, 1633–1674 (2018). https://doi.org/10.1007/s00028-018-0455-1
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DOI: https://doi.org/10.1007/s00028-018-0455-1