Planar Traveling Waves of Mono-Stable Reaction-Diffusion Equations

Abstract

This paper is concerned with planar traveling wavefronts of mono-stable reaction-diffusion equations in ${\mathbb{R}}^n$ ($n\ge 2$). We show that the large time behavior of the disturbed fronts can be controlled by two functions, which are the solutions of the specified nonlinear parabolic equations in ${\mathbb{R}}^{n-1}$, and the planar traveling fronts are asymptotically stable in $L^{\infty }({\mathbb{R}}^n)$ under ergodic perturbations, which include quasi-periodic and almost periodic ones as special cases.