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

 

Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable


Authors: Marcelo M. Cavalcanti, Irena Lasiecka and Daniel Toundykov
Journal: Trans. Amer. Math. Soc. 364 (2012), 5693-5713
MSC (2010): Primary 35L05; Secondary 93B07, 93D15
Published electronically: June 12, 2012
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Abstract: A stabilization/observability estimate and asymptotic energy decay rates are derived for a wave equation with nonlinear damping in a portion of the interior and Wentzell condition on the boundary: $ \partial _{\nu } u + u = \Delta _{T}u$. The dissipation does not affect a full collar of the boundary, thus leaving out a portion subjected to the high-order Wentzell condition.

Observability of wave equations with damping supported away from the Neumann boundary is known to be intrinsically more difficult than the corresponding Dirichlet problem because the uniform Lopatinskii condition is not satisfied by such a system. In the case of a Wentzell boundary, the situation is more difficult since the ``natural'' energy now includes the $ H^{1}$ Sobolev norm of the solution on the boundary. To establish uniform stability it is necessary not only to overcome the presence of the Neumann boundary operator, but also to establish an inverse-type coercivity estimate on the $ H^{1}$ trace norm of the solution. This goal is attained by constructing multipliers based on a refinement of nonradial vector fields employed for ``unobserved'' Neumann conditions. These multipliers, along with a suitable geometry (local convexity), allow reconstruction of the high-order part of the potential energy from the damping that is supported only in a far-off region of the domain.


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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

Irena Lasiecka
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904 – and – Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahram, 31261, Saudi Arabia
Email: il2v@virginia.edu

Daniel Toundykov
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588
Email: dtoundykov@math.unl.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05583-8
PII: S 0002-9947(2012)05583-8
Keywords: Wentzell, Ventcel, Venttsel, wave equation, nonlinear damping, localized damping, energy decay, multipliers
Received by editor(s): July 31, 2010
Published electronically: June 12, 2012
Additional Notes: The research of the first author was partially supported by the CNPq under Grant 300631/2003-0.
The research of the second author was partially supported by the National Science Foundation under Grant DMS-0606682 and by AFOSR Grant FA9550-09-1-0459
The research of the third author was partially supported by the National Science Foundation under Grant DMS-0908270
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.