Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity

This paper is concerned with traveling wave solutions of the equation partial derivative u/partial derivative t = J * u - u + f (u) on R x (0,infinity), where the dispersion kernel J is nonnegative and the nonlinearity f is monostable type. We show that there exists c* is an element of R such that for any c > c*, there is a nonincreasing traveling wave solution phi with phi(-infinity) = 1 and lim(xi ->infinity) phi(xi)e(lambda xi) = 1, where lambda =Lambda(1)(c) is the smallest positive solution to c lambda = integral(R)J(z)e(lambda z)dz - 1 + f'(0). Furthermore, the existence of traveling wave solutions with c = c* is also established. For c not equal 0, we further prove that the traveling wave solution is unique up to translation and is globally asymptotically stable in certain sense. (C) 2010 Elsevier Ltd. All rights reserved.