Existence and Nonexistence of Traveling Waves for a Nonlocal Monostable Equation

Abstract

We consider the nonlocal analogue of the Fisher-KPP equation
ut = 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∈ℝ_e–λy_dμ(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–λy_dμ(y) = +∞ for all positive constants λ.