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March  2009, 2(1): 67-94. doi: 10.3934/dcdss.2009.2.67

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

Moez Daoulatli 1, , Irena Lasiecka 2, and  Daniel Toundykov 3,

1. 

LAMSIN, ENIT, University of Tunis Elmanar, Tunisia
2. 

Kerchof Hall , P. O. Box 400137, University of Virginia, Charlottesville, VA 22904-4137, United States
3. 

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588

Received  January 2008 Revised  July 2008 Published  January 2009

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.
Keywords: decay rates., boundary damping, nonlinear dissipation, Wave equation.
Mathematics Subject Classification: Primary: 35L05, 35B40; Secondary: 35L70, 35B3.
Citation: Moez Daoulatli, Irena Lasiecka, Daniel Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 67-94. doi: 10.3934/dcdss.2009.2.67
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