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Stabilization of a Boussinesq system with localized damping

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Abstract

A family of Boussinesq systems was proposed by J. L. Bona, M. Chen, and J.-C. Saut to describe the two-way propagation of small amplitude gravity waves on the surface of water in a canal. Our work considers a class of these Boussinesq systems which couples two Benjamin-Bona-Mahony type equations posed on a bounded interval. We study the stabilization of the resulting system when a localized damping term acts in one equation only. By means of spectral analysis and eigenvectors expansion of solutions, we prove that the energy associated to the model converges to zero as time tends to infinity. Also, we address the problem of unique continuation property for the corresponding conservative system.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Sorin Micu

  2. Institute of Mathematical Statistics and Applied Mathematics, 70700, Bucharest, Romania

    Sorin Micu

  3. Institute of Mathematics, Federal University of Rio de Janeiro, UFRJ, P.O. Box 68530, CEP 21945-970, Rio de Janeiro, RJ, Brasil

    Ademir F. Pazoto

Authors
  1. Sorin Micu
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  2. Ademir F. Pazoto
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Corresponding author

Correspondence to Sorin Micu.

Additional information

Partially supported by Grant PN-II-ID-PCE-2011-3-0257 of the Romanian National Authority for Scientific Research, CNCS UEFISCDI (Romania) and by Grant MTM2011-29306 funded by MICINN (Spain).

Supported by CNPq (Brazil).

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Micu, S., Pazoto, A.F. Stabilization of a Boussinesq system with localized damping. JAMA 137, 291–337 (2019). https://doi.org/10.1007/s11854-018-0074-3

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  • DOI: https://doi.org/10.1007/s11854-018-0074-3

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