Exponential stability of the  wave equation withboundary  time-varying delay

Serge Nicaise; Cristina Pignotti; Julie Valein

doi:10.3934/dcdss.2011.4.693

Article Contents

2011, Volume 4, Issue 3: 693-722. Doi: 10.3934/dcdss.2011.4.693

This issue Previous Article An identification problem for a linear evolution equation in a Banach space and applications Next Article Linear evolution equations with strongly measurable families and application to the Dirac equation

Exponential stability of the wave equation with boundary time-varying delay

1.
Université de Valenciennes et du Hainaut Cambrésis, LAMAV and FR CNRS 2956, Le Mont Houy, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9
2.
Dipartimento di Matematica Pura e Applicata, Università di L'Aquila, Via Vetoio, Loc. Coppito, 67010 L'Aquila
3.
Institut Elie Cartan de Nancy, Université Henri Poincaré, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex

Received: March 2009

Revised: November 2009

Published: June 2011

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Abstract

We consider the wave equation with a time-varying delay term in the boundary condition in a bounded and smooth domain $\Omega\subset\RR^n.$ Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapunov functionals. Such analysis is also extended to a nonlinear version of the model.

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Mathematics Subject Classification: 35L05, 93D15.

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