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Imperfect Geometric Control and Overdamping for The Damped Wave Equation

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Abstract

We consider the damped wave equation on a manifold with imperfect geometric control. We show the sub-exponential energy decay estimate in (Christianson, J Funct Anal 258(3):1060–1065, 2010) is optimal in the case of one hyperbolic periodic geodesic. We show if the equation is overdamped, then the energy decays exponentially. Finally we show if the equation is overdamped but geometric control fails for one hyperbolic periodic geodesic, then nevertheless the energy decays exponentially.

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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Orsay (UMR 8628 CNRS), Université Paris-Sud, Bat. 425, 91405, Orsay Cedex, France

    Nicolas Burq

  2. Department of Mathematics, UNC-Chapel Hill, CB#3250 Phillips Hall, Chapel Hill, NC, 27599, USA

    Hans Christianson

Authors
  1. Nicolas Burq
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  2. Hans Christianson
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Correspondence to Hans Christianson.

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Communicated by Y. Kawahigashi

N.B. was supported in part by Agence Nationale de la Recherche project NOSEVOL, 2011 BS01019 01, and H.C. was supported in part by NSF Grant DMS-1059618.

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Burq, N., Christianson, H. Imperfect Geometric Control and Overdamping for The Damped Wave Equation. Commun. Math. Phys. 336, 101–130 (2015). https://doi.org/10.1007/s00220-014-2247-y

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  • DOI: https://doi.org/10.1007/s00220-014-2247-y

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