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

Serge Nicaise; Julie Valein

doi:10.3934/nhm.2007.2.425

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2007, Volume 2, Issue 3: 425-479. Doi: 10.3934/nhm.2007.2.425

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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

Received: September 30, 2006

Revised: April 30, 2007

Published: August 2007

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Abstract

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: 35L10.

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