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The role of the range of dispersal in a nonlocal Fisher-KPP equation: an asymptotic analysis
Communications in Contemporary Mathematics ( IF 1.278 ) Pub Date : 2020-06-22 , DOI: 10.1142/s0219199720500327
Julien Brasseur

In this paper, we study the asymptotic behavior as 𝜀0+ of solutions u𝜀 to the nonlocal stationary Fisher-KPP type equation 1𝜀mNJ𝜀(xy)(u𝜀(y)u𝜀(x))dy+u𝜀(x)(a(x)u𝜀(x))=0 in N, where 𝜀>0 and 0m<2. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution u𝜀 and that u𝜀a+ as 𝜀0+ where a+=max{0,a}. This generalizes the previously known results and answers an open question raised by Berestycki et al. Our method of proof is also of independent interest as it shows how to reduce this nonlocal problem to a local one. The sharpness of our assumptions is also briefly discussed.

更新日期:2020-08-09

 

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