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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result


Authors: M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano
Journal: Trans. Amer. Math. Soc. 361 (2009), 4561-4580
MSC (2000): Primary 32J15, 35L05, 47J35, 93D15
Published electronically: April 13, 2009
MathSciNet review: 2506419
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Abstract: This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by

$\displaystyle \left. \begin{array}{l} u_{tt} - \Delta_{\mathcal{M}}u+ a(x) g(u_... ...pace }\mathcal{M}\times \left] 0,\infty \right[ ,\smallskip \end{array} \right.$    

where $ \mathcal{M}\subset \mathbb{R}^3$ is a smooth oriented embedded compact surface without boundary. Denoting by $ \mathbf{g}$ the Riemannian metric induced on $ \mathcal{M}$ by $ \mathbb{R}^3$, we prove that for each $ \epsilon > 0$, there exist an open subset $ V \subset\mathcal M$ and a smooth function $ f:\mathcal M \rightarrow \mathbb{R}$ such that $ meas(V)\geq meas(\mathcal M)-\epsilon$, $ Hess f \approx \mathbf{g}$ on $ V$ and $ \underset{x\in V}\inf \vert\nabla f(x)\vert>0$.

In addition, we prove that if $ a(x) \geq a_0> 0$ on an open subset $ \mathcal{M}{\ast} \subset \mathcal M$ which contains $ \mathcal{M}\backslash V$ and if $ g$ is a monotonic increasing function such that $ k \vert s\vert \leq \vert g(s)\vert \leq K \vert s\vert$ for all $ \vert s\vert \geq 1$, then uniform and optimal decay rates of the energy hold.


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Additional Information

M. M. Cavalcanti
Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil

V. N. Domingos Cavalcanti
Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil

R. Fukuoka
Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil

J. A. Soriano
Affiliation: Department of Mathematics, State University of Maringá, 87020-900, Maringá, PR, Brazil

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04763-1
PII: S 0002-9947(09)04763-1
Keywords: Compact surfaces, wave equation, locally distributed damping.
Received by editor(s): April 26, 2007
Published electronically: April 13, 2009
Additional Notes: The research of the first author was partially supported by the CNPq Grant 300631/2003-0
The research of the second author was partially supported by the CNPq Grant 304895/2003-2
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.