# 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

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