Existence and Nonexistence of Traveling Waves for a Nonlocal Monostable Equation

Abstract

We consider the nonlocal analogue of the Fisher-KPP equation

where $\mu$ is a Borel-measure on $R$ with $\mu (R)=1$ and $f$ satisfies $f(0)=f(1)=0$ and $f>0$ in $(0,1)$. We do not assume that $\mu$ is absolutely continuous with respect to the Lebesgue measure. The equation may have a standing wave solution whose profile is a monotone but discontinuous function. We show that there is a constant $c_{\ast }$ such that it has a traveling wave solution with speed $c$ when $c\ge c_{\ast }$ while no traveling wave solution with speed $c$ when $c<c_{\ast }$, provided ${\int }_{y\in R}{e}^{–\lambda y}d\mu (y)<+\infty$ for some positive constant $\lambda$. In order to prove it, we modify a recursive method for abstract monotone discrete dynamical systems by Weinberger. We note that the monotone semiflow generated by the equation is not compact with respect to the compact-open topology. We also show that it has no traveling wave solution, provided $f^{\prime }(0)>0$ and  ${\int }_{y\in R}{e}^{–\lambda y}d\mu (y)=+\infty$ for all positive constants $\lambda$.