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

Abstract

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