Existence and nonexistence of traveling waves for a nonlocal monostable equation

By Hiroki Yagisita. 2008

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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 0 and ∫y∈Re-λydµ(y) = + ∞ for all positive constants λ.

Subfields: Applied Mathematics, Control and Systems Engineering, Public Health, Environmental and Occupational Health

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