Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations

Abstract

The existence, uniqueness, and global exponential stability of traveling wave solutions of a class of nonlinear and nonlocal evolution equations are established. It is assumed that there are two stable equilibria so that a traveling wave is a solution that connects them. A basic assumption is the comparison principle: a smaller initial value produces a smaller solution. When applied to differential equations or integro-differential equations, the result recovers and/or complements a number of existing ones.