Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions

Moez Daoulatli; Irena Lasiecka; Daniel Toundykov

doi:10.3934/dcdss.2009.2.67

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2009, Volume 2, Issue 1: 67-94. Doi: 10.3934/dcdss.2009.2.67

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Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions

1.
LAMSIN, ENIT, University of Tunis Elmanar
2.
Kerchof Hall , P. O. Box 400137, University of Virginia, Charlottesville, VA 22904-4137
3.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received: December 31, 2007

Revised: June 30, 2008

Published: February 2009

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Abstract

This paper studies a wave equation on a bounded domain in $\bbR^d$ with nonlinear dissipation which is localized on a subset of the boundary. The damping is modeled by a continuous monotone function without the usual growth restrictions imposed at the origin and infinity. Under the assumption that the observability inequality is satisfied by the solution of the associated linear problem, the asymptotic decay rates of the energy functional are obtained by reducing the nonlinear PDE problem to a linear PDE and a nonlinear ODE. This approach offers a generalized framework which incorporates the results on energy decay that appeared previously in the literature; the method accommodates systems with variable coefficients in the principal elliptic part, and allows to dispense with linear restrictions on the growth of the dissipative feedback map.

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Mathematics Subject Classification: Primary: 35L05, 35B40; Secondary: 35L70, 35B35.

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