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This paper is concerned with the critical sharp traveling wave for doublynonlinear diffusion equation with time delay, where the doubly nonlineardegenerate 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 toadmit 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 withthe minimal wave speed $c^*(m,p,r)$ is monotonically increasing, where theminimal 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 $\frac{1}{p-1}<m< \frac{p}{p-1}$, andpiecewise smooth for $m\ge \frac{p}{p-1}$. Our results indicate that time delayslows down the minimal traveling wave speed for the doubly nonlinear degeneratediffusion equations. The approach adopted for proof is the phase transformmethod combining the variational method. The main technical issue for the proofis to overcome the obstacle caused by the doubly nonlinear degeneratediffusion.

Critical Sharp Front for Doubly Nonlinear Degenerate Diffusion Equations with Time Delay