# American Institute of Mathematical Sciences

November  2014, 34(11): 4371-4388. doi: 10.3934/dcds.2014.34.4371

## Polynomial stabilization of some dissipative hyperbolic systems

 1 UR Analyse et Contrôle des Edp (05/UR/15-01), Département de Mathématiques, Faculté des Sciences de Monastir, Université de Monastir, 5019 Monastir, Tunisia 2 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1 3 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

Received  September 2013 Revised  January 2014 Published  May 2014

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.
Citation: Kais Ammari, Eduard Feireisl, Serge Nicaise. Polynomial stabilization of some dissipative hyperbolic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4371-4388. doi: 10.3934/dcds.2014.34.4371
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