Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Polynomial stabilization of some dissipative hyperbolic systems

Pages: 4371 - 4388, Volume 34, Issue 11, November 2014      doi:10.3934/dcds.2014.34.4371

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Kais Ammari - UR Analyse et Contrôle des Edp (05/UR/15-01), Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, 5019 Monastir, Tunisia (email)
Eduard Feireisl - Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic (email)
Serge Nicaise - 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, France (email)

Abstract: We study the problem of stabilization for the acoustic system with a spatially distributed damping. Imposing various hypotheses on the structural properties of the damping term, we identify either exponential or polynomial decay of solutions with growing time. Exponential decay rate is shown by means of a time domain approach, reducing the problem to an observability inequality to be verified for solutions of the associated conservative problem. In addition, we show a polynomial stabilization result, where the proof uses a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent.

Keywords:  Exponential stability, polynomial stability, observability inequality, resolvent estimate, dissipative hyberbolic system, acoustic equation.
Mathematics Subject Classification:  Primary: 35L04, 93B07; Secondary: 93B52, 74H55.

Received: September 2013;      Revised: January 2014;      Available Online: May 2014.