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

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.

References in corpus (4)

• Non-monotone travelling waves in a single species reaction-diffusion equation with delay
• Global continuation of monotone wavefronts
• Travelling waves for a non-monotone bistable equation with delay: existence and oscillations
• Variational Approach of Critical Sharp Front Speeds in Density-dependent Diffusion Model with Time Delay

Cited by in corpus (1)

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