Can Decaying Particles Raise the Upperbound on the Peccei-Quinn Scale?

We have reexamine the effect of entropy production on the cosmic axion density and find that the Peccei-Quinn scale $F_a$ larger than about $10^{15}$~GeV is not allowed even if large entropy is produced by the decays of coherent oscillations or non-relativistic massive particles. We stress that this result is independent of the details of models for the decaying particles.