TY - JOUR
T1 - The Bramson delay in the non-local Fisher-KPP equation
AU - Bouin, Emeric
AU - Henderson, Christopher
AU - Ryzhik, Lenya
N1 - Funding Information:
Acknowledgments The authors thank Nicolas Champagnat for the reference [8]. EB was supported by “INRIA Programme Explorateur”. LR was supported by NSF grants DMS-1311903 and DMS-1613603. Part of this work was performed within the framework of the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d'Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). In addition, CH has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (grant agreement No 639638) and was partially supported by the NSF RTG grant DMS-1246999 and DMS-1907853.
Publisher Copyright:
© 2019
PY - 2020/1/1
Y1 - 2020/1/1
N2 - We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either 2t−(3/2)logt+O(1), as in the local case, or 2t−O(tβ) for some explicit β∈(0,1). Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any β∈(0,1), examples of Fisher-KPP type non-linearities fβ such that the front for the local Fisher-KPP equation with reaction term fβ is at 2t−O(tβ).
AB - We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either 2t−(3/2)logt+O(1), as in the local case, or 2t−O(tβ) for some explicit β∈(0,1). Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any β∈(0,1), examples of Fisher-KPP type non-linearities fβ such that the front for the local Fisher-KPP equation with reaction term fβ is at 2t−O(tβ).
KW - Logarithmic delay
KW - Parabolic Harnack inequality
KW - Reaction-diffusion equations
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U2 - 10.1016/j.anihpc.2019.07.001
DO - 10.1016/j.anihpc.2019.07.001
M3 - Article
AN - SCOPUS:85075362290
SN - 0294-1449
VL - 37
SP - 51
EP - 77
JO - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
JF - Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
IS - 1
ER -