Traveling wave solutions of advection–diffusion equations with nonlinear diffusion

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∈]c⁎,+∞[, where c⁎>0 is explicitly computed but may not be optimal. We also prove that a free boundary hypersurface 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.