Front motion in viscous conservation laws with stiff source terms

Abstract

We study a scalar, bi-stable reaction-diffusion-convection equation in $\mathbb{R}^N$. With a hyperbolic scaling, we reduce it to a singularly perturbed equation. In the singular limit, solutions converge to functions which take on only two different values in bulk regions. We also derive an equation describing the motion of the interface between the two bulk regions. To the lowest order, the normal speed $s(\nu)$ of the interface depends only on the unit normal vector $\nu$, where the wave speed $s(\nu)$ is determined by nonlinear reaction and convection terms. When the convection term is even and the reaction term is odd in their argument, the wave speed $s(\nu)$ vanishes identically for all directions $\nu$. In this situation, a parabolic scaling reduces the equation to another singularly perturbed equation for which we also establish convergence of solutions in the singular limit. The singular limit dynamics is governed by an anisotropic mean curvature flow, in which the anisotropy comes from the convection term. Our method of proof for convergence consists of constructing approximate solutions of any order and using the comparison principle.