Critical sharp front for doubly nonlinear degenerate diffusion equations with time delay

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