Polynomial decay for a hyperbolic–parabolic coupled system☆☆The work is supported by the US National Science Foundation grant NSF-DMS-0104096, the Grant BFM2002-03345 of the Spanish MCYT, the FANEDD of China (Project No. 200119), the EU TMR Projects “Homogenization and Multiple Scales” and “Smart Systems”, the NSF of China under grant 10371084, and the Project-sponsored by SRF for ROCS, SEM of China. The first and last authors would like to thank the Laboratoire J.-A. Dieudonné at the Université de Nice, and particularly Professors Y. Brenier, D. Chenais and G. Lebeau, for hospitality during their visit. The third author also thanks Professors N. Burq, I. Mundet and V. Munõs for fruitful discussions.