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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation


Authors: Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti and Flávio A. F. Nascimento
Journal: Proc. Amer. Math. Soc. 141 (2013), 3183-3193
MSC (2010): Primary 35L05, 34Dxx, 35A27
Published electronically: May 29, 2013
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Abstract: We discuss the asymptotic stability of the wave equation on a compact Riemannian manifold $ (M, \bf g)$ subject to locally distributed viscoelastic effects on a subset $ \omega \subset M$. Assuming that the well-known geometric control condition $ (\omega , T_0)$ holds and supposing that the relaxation function is bounded by a function that decays exponentially to zero, we show that the solutions of the corresponding partial viscoelastic model decay exponentially to zero. We give a new geometric proof extending the prior results in the literature from the Euclidean setting to compact Riemannian manifolds (with or without boundary).


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Additional Information

Marcelo M. Cavalcanti
Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil
Email: mmcavalcanti@uem.br

Valéria N. Domingos Cavalcanti
Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil
Email: vndcavalcanti@uem.br

Flávio A. F. Nascimento
Affiliation: Department of Mathematics, State University of Ceará-FAFIDAM, 62930-000, Limoeiro do Norte, CE, Brazil
Email: flavio.falcao@uece.br

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11869-1
PII: S 0002-9939(2013)11869-1
Keywords: Wave equation, compact Riemannian manifold, viscoelastic distributed damping
Received by editor(s): November 28, 2011
Published electronically: May 29, 2013
Additional Notes: Research of the first author was partially supported by the CNPq Grant 300631/2003-0
Research of the second author was partially supported by the CNPq Grant 304895/2003-2
The third author, a doctorate student at the State University of Maringá, was partially supported by a grant of CNPq, Brazil
Communicated by: James E. Colliander
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.