Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity.

*(English)*Zbl 1346.35109The authors perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. They introduce two scalings. The first part of the paper is devoted to the long time/long range rescaling of the Fisher-KPP equation describing population dispersion with fractional Laplacian. This rescaling is based on previous results of Cabre et al. about the exponential speed of propagation of the population. The authors show that the exponential propagation is derived only from the form of the solution at the initial layer. They further show that the only role of the fractional Laplacian in determining this speed is at the initial layer where it determines the thickness of the tails of the solutions. The resulting convergence is locally uniformly in the appropriate spaces. The authors notice that at the limit, when the small parameter vanishes, the fractional Laplacian does not have any impact on the dynamics of the limit population and the dynamics are determined only by the reaction term. In the second part, the authors show that such rescaling is also possible for models with nonlocal reaction terms. For instance, in the case of selection-mutation models in which it is considered only a dependence on the mean-field competition term. Further, they consider a second rescaling where it is assumed that the diffusion steps are small. Using a WKB ansatz, they obtain a Hamilton-Jacobi equation in the limit which describes the asymptotic dynamics of the solutions, similarly to the case of selection-mutation models with a classical Laplace term or an integral kernel with thin tails. Diffusion with small steps and long time is also considered. They extend these results to the multidimensional case.

Reviewer: Mariano Rodríguez Ricard (La Habana)

##### MSC:

35K57 | Reaction-diffusion equations |

35B40 | Asymptotic behavior of solutions to PDEs |

35B25 | Singular perturbations in context of PDEs |

47G20 | Integro-differential operators |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |

35R11 | Fractional partial differential equations |

##### Keywords:

asymptotic behavior; Fisher-KPP equation; fractional Laplacian; Hamilton-Jacobi equation; local and nonlocal competition
PDF
BibTeX
XML
Cite

\textit{S. Méléard} and \textit{S. Mirrahimi}, Commun. Partial Differ. Equations 40, No. 5, 957--993 (2015; Zbl 1346.35109)

Full Text:
DOI

**OpenURL**

##### References:

[1] | DOI: 10.1007/s11538-007-9220-2 · Zbl 1296.92195 |

[2] | Barles G., Solutions de viscosité des Équations de Hamilton-Jacobi (1994) |

[3] | DOI: 10.1215/S0012-7094-90-06132-0 · Zbl 0749.35015 |

[4] | DOI: 10.1016/j.anihpc.2007.02.007 · Zbl 1155.45004 |

[5] | Barles G., Methods Appl. Anal. 16 pp 321– (2009) |

[6] | DOI: 10.1137/0326063 · Zbl 0674.49027 |

[7] | Berestycki H., Discrete Contin. Dyn. Syst. Ser. S pp 1– (2011) |

[8] | DOI: 10.1016/j.na.2009.04.059 · Zbl 1226.60041 |

[9] | DOI: 10.1090/S0002-9947-2013-05629-2 · Zbl 1281.47032 |

[10] | DOI: 10.1016/j.crma.2012.10.007 · Zbl 1253.35198 |

[11] | DOI: 10.1016/j.crma.2009.10.012 · Zbl 1182.35072 |

[12] | DOI: 10.1007/s00220-013-1682-5 · Zbl 1307.35310 |

[13] | DOI: 10.1080/03605302.2012.718024 · Zbl 1263.35142 |

[14] | DOI: 10.1090/S0273-0979-1992-00266-5 · Zbl 0755.35015 |

[15] | DOI: 10.1016/j.tpb.2004.12.003 · Zbl 1072.92035 |

[16] | Engler H., Int. J. Differ. Equ. (2010) |

[17] | DOI: 10.1512/iumj.1989.38.38007 · Zbl 0692.35014 |

[18] | Freidlin M., Functional Integration and Partial Differential Equations (1985) · Zbl 0568.60057 |

[19] | DOI: 10.1214/aop/1176992901 · Zbl 0576.60070 |

[20] | DOI: 10.1016/0022-5193(75)90011-9 |

[21] | DOI: 10.1016/j.spa.2005.11.006 · Zbl 1104.60030 |

[22] | DOI: 10.1007/s00285-011-0478-5 · Zbl 1311.92136 |

[23] | DOI: 10.1137/10080693X · Zbl 1232.47058 |

[24] | DOI: 10.1512/iumj.2008.57.3398 · Zbl 1172.35005 |

[25] | Sato K., Lévy processes and Infinitely Divisible Distributions (1999) · Zbl 0973.60001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.