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

Abstract

We consider traveling fronts to the nonlocal bistable equation

where $\mu$ is a Borel-measure on $R$ with $\mu (R)=1$ and $f$ satisfies $f(0)=f(1)=0$, $f<0$ in $(0,\alpha )$ and $f>0$ in $(\alpha ,1)$ for some constant $\alpha \in (0,1)$. We do not assume that $\mu$ is absolutely continuous with respect to the Lebesgue measure. We show that there are a constant $c$ and a monotone function $\varphi$ with $\varphi (–\infty )=0$ and $\varphi (+\infty )=1$ such that $u(t,x):=\varphi (x+ct)$ is a solution to the equation, provided $f^{\prime }(\alpha )>0$. In order to prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.