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Singular limits for reaction-diffusion equations with fractional Laplacian and local or nonlocal nonlinearity. (English) Zbl 1346.35109
The 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.

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
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