Traveling Wave Solutions of Advection-Diffusion Equations with Nonlinear Diffusion

TL;DR

This paper investigates the existence and properties of traveling wave solutions in nonlinear degenerate diffusion equations with shear flow, including free boundary behavior and asymptotic linearity at infinity.

Contribution

It establishes existence conditions for traveling waves with explicit minimal speeds and analyzes the free boundary and asymptotic properties of solutions.

Findings

Existence of traveling wave solutions for speeds above a computed threshold.

Characterization of free boundary as a Lipschitz continuous graph.

Solutions are asymptotically planar and linear at infinity.

Abstract

We study the existence of particular traveling wave solutions of a nonlinear parabolic degenerate diffusion equation with a shear flow. Under some assumptions we prove that such solutions exist at least for propagation speeds c {\in}]c*, +{\infty}, where c* > 0 is explicitly computed but may not be optimal. We also prove that a free boundary hy- persurface separates a region where u = 0 and a region where u > 0, and that this free boundary can be globally parametrized as a Lipschitz continuous graph under some additional non-degeneracy hypothesis; we investigate solutions which are, in the region u > 0, planar and linear at infinity in the propagation direction, with slope equal to the propagation speed.