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.
Cite this article
Hiroki Yagisita, Existence and Nonexistence of Traveling Waves for a Nonlocal Monostable Equation. Publ. Res. Inst. Math. Sci. 45 (2009), no. 4, pp. 925–953DOI 10.2977/PRIMS/1260476648