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From resolvent estimates to damped waves

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Abstract

In this paper, we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the undamped set. We show that if replacing the damping term with a higher-order complex absorbing potential gives an operator enjoying polynomial resolvent bounds on the real axis, then the “resolvent” associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding non-compact models.

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Authors and Affiliations

  1. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    Hans Christianson

  2. LAGA - CNRS (UMR 7539), Université Paris 13, 93430, Villetaneuse, France

    Emmanuel Schenck

  3. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    András Vasy

  4. Department of Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Jared Wunsch

Authors
  1. Hans Christianson
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  2. Emmanuel Schenck
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  3. András Vasy
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  4. Jared Wunsch
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Corresponding author

Correspondence to Hans Christianson.

Additional information

H. C. was partially supported by NSF grant DMS-0900524.

A. V. was partially supported by NSF grant DMS-1068742.

J. W. was partially supported by NSF grant DMS-1001463.

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Christianson, H., Schenck, E., Vasy, A. et al. From resolvent estimates to damped waves. JAMA 122, 143–162 (2014). https://doi.org/10.1007/s11854-014-0006-9

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  • DOI: https://doi.org/10.1007/s11854-014-0006-9

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