  This paper is concerned with the critical sharp traveling wave for doubly
nonlinear diffusion equation with time delay, where the doubly nonlinear
degenerate diffusion is defined by $\Big(\big|(u^m)_x\big|^{p-2}(u^m)_x\Big)_x$
with $m>0$ and $p>1$. The doubly nonlinear diffusion equation is proved to
admit a unique sharp type traveling wave for the degenerate case $m(p-1)>1$,
the so-called slow-diffusion case. This sharp traveling wave associated with
the minimal wave speed $c^*(m,p,r)$ is monotonically increasing, where the
minimal wave speed satisfies $c^*(m,p,r)<c^*(m,p,0)$ for any time delay $r>0$.
The sharp front is $C^1$-smooth for $\frac{1}{p-1}<m< \frac{p}{p-1}$, and
piecewise smooth for $m\ge \frac{p}{p-1}$. Our results indicate that time delay
slows down the minimal traveling wave speed for the doubly nonlinear degenerate
diffusion equations. The approach adopted for proof is the phase transform
method combining the variational method. The main technical issue for the proof
is to overcome the obstacle caused by the doubly nonlinear degenerate
diffusion.
