Asymptotic formula for the speed of traveling wave solutions to Allen-Cahn equation

Abstract

In this paper, we investigate the asymptotic formula of the speed of the traveling wave solutions to the Allen-Cahn equations under linear perturbations with the usual and fractional Laplacians. The key ingredient is to estimate the traveling speeds in terms of the potential function. By estimating the traveling speed in terms of the nonlinear potential function, we can get the uniform estimates of solutions and uniform decay of derivatives of solutions at infinity, which will give us the asymptotic formula for the traveling speeds.