Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-A sharp result
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- by M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka and J. A. Soriano PDF
- Trans. Amer. Math. Soc. 361 (2009), 4561-4580 Request permission
Abstract:
This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by \begin{equation} \left . \begin {array}{l} u_{tt} - \Delta _{\mathcal {M}}u+ a(x) g(u_{t})=0 \; \text {on \thinspace }\mathcal {M}\times \left ] 0,\infty \right [ , \end{array} \right . \nonumber \end{equation} 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 $\underset {x\in V}\inf |\nabla f(x)|>0$.
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 $k |s| \leq |g(s)| \leq K |s|$ for all $|s| \geq 1$, 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
- MR Author ID: 622908
- 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
- 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 - © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 4561-4580
- MSC (2000): Primary 32J15, 35L05, 47J35, 93D15
- DOI: https://doi.org/10.1090/S0002-9947-09-04763-1
- MathSciNet review: 2506419