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

### Hiroki Yagisita

Kyoto University, Japan

## Abstract

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. 955–979

DOI 10.2977/PRIMS/1260476649