Geometrically constrained stabilization of wave equations with Wentzell boundary conditions

Abstract

Uniform stabilization of wave equation subject to second-order boundary conditions is considered in this article. Both dynamic (Wentzell) and static (with higher derivatives in space only) boundary conditions are discussed. In contrast to the classical wave equation where stabilization can be achieved by applying boundary velocity feedback, for a Wentzell-type problem boundary damping alone does not cause the energy to decay uniformly to zero. This is the case for both dynamic and static second-order conditions. In order to achieve uniform decay rates of the associated energy, it is necessary to dissipate part of the collar near the boundary. It will be shown how a combination of partially localized boundary feedback and partially localized collar feedback leads to uniform decay rates that are described by a nonlinear differential equation. This goal is attained by combining techniques used for stabilization of ‘unobserved’ Neumann conditions with differential geometry techniques effective for stabilization on compact manifolds. These lead to a construction of special non-radial multipliers which are geometry dependent and 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.

Keywords:

• Wentzell
• Ventcel
• Venttsel
• wave equation
• nonlinear damping
• localized damping
• energy decay
• multipliers

AMS Subject Classifications::

• Primary: 35L05
• Secondary: 35L53, 93B07, 93D15

Acknowledgements

The authors express their gratitude to the anonymous referees for the detailed corrections and comments which have helped improve this work. The research of M.M. Cavalcanti was partially supported by the CNPq under Grant 300631/2003-0. The research of I. Lasiecka was partially supported by the National Science Foundation under Grant DMS-0606682 and by AFOSR Grant FA9550-09-1-0459. The research of D. Toundykov was partially supported by the National Science Foundation under Grant DMS-0908270.