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

Cavalcanti, Marcelo; Lasiecka, Irena; Toundykov, Daniel

doi:10.1090/S0002-9947-2012-05583-8

Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable
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by Marcelo M. Cavalcanti, Irena Lasiecka and Daniel Toundykov PDF

Trans. Amer. Math. Soc. 364 (2012), 5693-5713 Request permission

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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