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

MathSciNet review:
3068971

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Abstract | References | Similar Articles | Additional Information

Abstract: We discuss the asymptotic stability of the wave equation on a compact Riemannian manifold subject to locally distributed viscoelastic effects on a subset . Assuming that the well-known geometric control condition 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

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.