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 * 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 ∈ (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 Φ with Φ(-∞) = 0 and Φ(+oo) = 1 such that u(t, x) := Φ(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. © 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.