  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. Expo- nential 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.
