  We consider the Cauchy problem for two prototypes of flux-saturated diffusion
equations. In arbitrary space dimension, we give an optimal condition on the
growth of the initial datum which discriminates between occurrence or
nonoccurrence of a waiting time phenomenon. We also prove optimal upper bounds
on the waiting time. Our argument is based on the introduction of suitable
families of subsolutions and on a comparison result for a general class of
flux-saturated diffusion equations.
