Existence and Nonexistence of Traveling Waves for a Nonlocal Monostable Equation

We consider the nonlocal analogue of the Fisher-KPP equation u_t = 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 and f > 0 in (0, 1) . We do not assume that μ 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 ∫ y∈ ℝe–λydμ(y) < +∞ for some positive constant λ . 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 ∫y∈ ℝ e–λydμ(y) = +∞ for all positive constants λ .