  For the classical reaction diffusion equation, the priori speed of fronts is
determined exactly in the pioneering paper (R.D. Benguria and M.C. Depassier,
{\em Commun. Math. Phys.} 175:221--227, 1996) by variational characterization
method. In this paper, we model the dispersal process using a density-dependent
diffusion equation with time delay. We show the existence and uniqueness of
sharp critical fronts, where the sharp critical front is $C^1$-smooth when the
diffusion degeneracy is weaker with $1<m<2$, and the sharp critical front is
non-$C^1$-smooth (piecewise smooth) when the diffusion degeneracy is stronger
with $m\ge 2$. We give a new variational approach for the critical wave speed
and investigate how the time delay affects the propagation mechanism of fronts.
Our results provide some interesting insight into the dynamics of critical
traveling wave.
