Existence and Nonexistence of Traveling Waves for a Nonlocal Monostable Equation

We consider the nonlocal analogue of the Fisher-KPP equation u(t) = mu * u - u + f(u), 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(*) such that it has a traveling wave solution with speed c when c >= c(*) while no traveling wave solution with speed c when c < c(*) provided integral(y is an element of R) e(-lambda y) d mu(y) < +infinity 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' (0) > 0 and integral(y is an element of R) e(-lambda y) d mu(y) = +infinity for all positive constants lambda.