Asymptotic stability of the wave equation on compact surfaces and   locally distributed damping - A sharp result
  This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping. We study the case where the damping is effective in a well-chosen subset of arbitrarily small measure. 
M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A. Soriano