Existence, Uniqueness, And Asymptotic Stability Of Traveling Waves In Nonlocal Evolution Equations

. 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 tr aveling 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. Key Words. Traveling wave, asymptotic stability, comparison principle, nonlocal evolution. AMS subject classifications (1991). 35B05, 35B40, 35A05, 35R99. 1 Introduction. In this paper, we are concerned with a one space dimensional evolution equation u t (x; t) = A[u(\Delta; t)](x); x 2 IR; t ? 0; (1.1) where A is a nonlinear operator which is independent of the time t, maps functions of space variable \Delta to functions of x, and, via (1.1), generates a semi-group on the Ban..