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September  2007, 2(3): 425-479. doi: 10.3934/nhm.2007.2.425

Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks

Serge Nicaise 1, and  Julie Valein 1,

1. 

Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9, France

Received  October 2006 Revised  May 2007 Published  June 2007

In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.
Keywords: wave equation, stabilization, delay.
Mathematics Subject Classification: Primary: 35L05, 93D15; Secondary: 35L1.
Citation: Serge Nicaise, Julie Valein. Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks & Heterogeneous Media, 2007, 2 (3) : 425-479. doi: 10.3934/nhm.2007.2.425
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