Existence of Traveling Wave Solutions for a Nonlocal Bistable Equation: An Abstract Approach

We consider traveling fronts to the nonlocal bistable equation u(t) = mu * u - u + f(u), where mu is a Borel-measure on R with mu(R) = 1 and f satisfies f (0) = f (1) = 0, f < 0 in (0, alpha) and f > 0 in (alpha, 1) for some constant alpha is an element of (0, 1). We do not assume that mu is absolutely continuous with respect to the Lebesgue measure. We show that there are a constant c and a monotone function phi with phi(-infinity) = 0 and phi(+infinity) = 1 such that u(t, x) := phi(x+ct) is a solution to the equation, provided f ''(alpha) > 0. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.