Asymptotic stability of the wave equation on compact manifolds and locally distributed viscoelastic dissipation
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- by Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti and Flávio A. F. Nascimento PDF
- Proc. Amer. Math. Soc. 141 (2013), 3183-3193 Request permission
Erratum: Proc. Amer. Math. Soc. 145 (2017), 4097-4097.
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).References
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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
- MR Author ID: 622908
- Email: vndcavalcanti@uem.br
- Flávio A. F. Nascimento
- Affiliation: Department of Mathematics, State University of Ceará-FAFIDAM, 62930-000, Limoeiro do Norte, CE, Brazil
- MR Author ID: 993402
- Email: flavio.falcao@uece.br
- 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
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3183-3193
- MSC (2010): Primary 35L05, 34Dxx, 35A27
- DOI: https://doi.org/10.1090/S0002-9939-2013-11869-1
- MathSciNet review: 3068971