Henri Berestycki et al 2009 Nonlinearity 22 2813 doi:10.1088/0951-7715/22/12/002
The non-local Fisher–KPP equation: travelling waves and steady states
Henri Berestycki1, Grégoire Nadin2, Benoit Perthame3,4 and Lenya Ryzhik5
Show affiliationsRecommended by J-P Eckmann
We consider the Fisher–KPP equation with a non-local saturation effect defined through an interaction kernel 
(x) and investigate the possible differences with the standard Fisher–KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transform 
is positive or if the length σ of the non-local interaction is short enough, then the only steady states are u ≡ 0 and u ≡ 1. Next, we study existence of the travelling waves. We prove that this equation admits travelling wave solutions that connect u = 0 to an unknown positive steady state u∞(x), for all speeds c ≥ c*. The travelling wave connects to the standard state u∞(x) ≡ 1 under the aforementioned conditions: 
or σ is sufficiently small. However, the wave is not monotonic for σ large.
- PACS
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02.30.Cj Measure and integration
- MSC
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30C40 Kernel functions and applications
45A05 Linear integral equations
45E05 Integral equations with kernels of Cauchy type (See also 35J15)
- Subjects
- Dates
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Issue 12 (December 2009)
Received 20 March 2009, in final form 27 August 2009
Published 30 October 2009
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The non-local Fisher–KPP equation: travelling waves and steady states
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