We consider traveling fronts to the nonlocal bistable equation
ut = 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(+∞) = 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.
Cite this article
Hiroki Yagisita, Existence of Traveling Wave Solutions for a Nonlocal Bistable Equation: An Abstract Approach. Publ. Res. Inst. Math. Sci. 45 (2009), no. 4, pp. 955–979DOI 10.2977/PRIMS/1260476649