Variational Approach of Critical Sharp Front Speeds in Density-dependent Diffusion Model with Time Delay

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.