A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

Issue		ESAIM: COCV Volume 14, Number 3, July-September 2008


Page(s)		561 - 574
DOI		10.1051/cocv:2007066
Published online		21 December 2007

ESAIM: COCV 14 (2008) 561-574
DOI: 10.1051/cocv:2007066

A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

Louis Tebou

Department of Mathematics, Florida International University, Miami FL 33199, USA; teboul@fiu.edu

Received August 18, 2006. Published online December 21, 2007.

Abstract
First, we consider a semilinear hyperbolic equation with a locally distributed damping in a bounded domain. The damping is located on a neighborhood of a suitable portion of the boundary. Using a Carleman estimate [Duyckaerts, Zhang and Zuazua, Ann. Inst. H. Poincaré Anal. Non Linéaire (to appear); Fu, Yong and Zhang, SIAM J. Contr. Opt. 46 (2007) 1578-1614], we prove that the energy of this system decays exponentially to zero as the time variable goes to infinity. Second, relying on another Carleman estimate [Ruiz, J. Math. Pures Appl. 71 (1992) 455-467], we address the same type of problem in an exterior domain for a locally damped semilinear wave equation. For both problems, our method of proof is constructive, and much simpler than those found in the literature. In particular, we improve in some way on earlier results by Dafermos, Haraux, Nakao, Slemrod and Zuazua.

Mathematics Subject Classification. 93D15, 35L05, 35L70

Key words: Hyperbolic equation, exponential decay, localized damping, Carleman estimates

© EDP Sciences, SMAI 2007

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