The Threshold between Effective and Noneffective Damping for Semilinear   Waves
  In this paper we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with scale-invariant damping. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. We extend our results to a model with polynomial speed of propagation and to a model with an exponential speed of propagation. 
Marcello D'Abbicco
math.AP