%0 Journal Article %1 berestycki2009nonlocal %A Berestycki, Henri %A Nadin, Gregoire %A Perthame, Benoit %A Ryzhik, Lenya %D 2009 %J Nonlinearity %K Fisher-KPP density_dependence reaction-diffusion travelling_wave %N 12 %P 2813-2844 %T The non-local Fisher-KPP equation: travelling waves and steady states %U http://stacks.iop.org/0951-7715/22/2813 %V 22 %X We consider the Fisher-KPP equation with a non-local saturation effect defined through an interaction kernel ph(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 \\hat\\phi(\\xi) is positive or if the length s 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: \\hat\\phi(\\xi)$>$0 or s is sufficiently small. However, the wave is not monotonic for s large.

@article{berestycki2009nonlocal, abstract = {We consider the Fisher-KPP equation with a non-local saturation effect defined through an interaction kernel ph(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 \\hat\\phi(\\xi) is positive or if the length s 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: \\hat\\phi(\\xi)$>$0 or s is sufficiently small. However, the wave is not monotonic for s large.}, added-at = {2009-11-12T03:46:24.000+0100}, author = {Berestycki, Henri and Nadin, Gregoire and Perthame, Benoit and Ryzhik, Lenya}, biburl = {https://www.bibsonomy.org/bibtex/292c72f7d43bda2b0bc081a0f38467a7b/peter.ralph}, interhash = {da8c48d772ed97d3accec9036a9f9ae3}, intrahash = {92c72f7d43bda2b0bc081a0f38467a7b}, journal = {Nonlinearity}, keywords = {Fisher-KPP density_dependence reaction-diffusion travelling_wave}, number = 12, pages = {2813-2844}, timestamp = {2023-03-20T18:26:25.000+0100}, title = {The non-local {Fisher}-{KPP} equation: travelling waves and steady states}, url = {http://stacks.iop.org/0951-7715/22/2813}, volume = 22, year = 2009 }