Sharp, Smooth, and Oscillatory Traveling Waves of Degenerate Diffusion Equation with Delay

TL;DR

This paper investigates a degenerate diffusion equation with delay, revealing the existence of sharp, smooth, and oscillatory traveling waves, and introduces new methods to analyze these complex wave phenomena.

Contribution

It develops a novel technique to prove the existence of various types of traveling waves in a degenerate diffusion equation with delay, addressing challenges not tackled by existing methods.

Findings

Existence of sharp and smooth traveling waves.

Discovery of sharp-oscillating waves with non-decaying oscillations.

Oscillatory properties for large wave speeds and delays.

Abstract

We consider the non-monotone degenerate diffusion equation with time delay. Different from the linear diffusion equation, the degenerate equation allows for semi-compactly supported traveling waves. In particular, we discover sharp-oscillating waves with sharp edges and non-decaying oscillations. The degenerate diffusion and the effect of time delay cause us essential difficulties. We show the existence for both sharp and smooth traveling wave solutions. Furthermore, we prove the oscillating properties of the waves for large wave speeds and large time delay. Since the existing approaches are not applicable, we develop a new technique to show the existence of the sharp, smooth and oscillatory traveling waves.