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

TL;DR

This paper studies the existence, uniqueness, and properties of critical sharp traveling waves in doubly nonlinear degenerate diffusion equations with time delay, revealing how delay affects wave speed and smoothness.

Contribution

It establishes the existence and uniqueness of sharp traveling waves for the slow-diffusion case and analyzes how time delay influences wave speed and regularity.

Findings

Unique sharp traveling wave exists for $m(p-1)>1$

Time delay decreases the minimal wave speed

Sharp front smoothness depends on parameters

Abstract

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.