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

TL;DR

This paper analyzes the asymptotic behavior of reaction-diffusion models with fractional Laplacians, revealing how fractional diffusion influences propagation speed and solution structure, especially under different rescaling regimes and in multiple dimensions.

Contribution

It introduces new rescaling techniques for fractional reaction-diffusion equations, deriving Hamilton-Jacobi limits and extending results to multidimensional settings.

Findings

Fractional Laplacian determines initial tail thickness and propagation speed.

Rescaling with small diffusion steps leads to Hamilton-Jacobi equations.

Results extend to multidimensional models.

Abstract

We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP equation. This rescaling is based on the exponential speed of propagation of the population. In particular we 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. Next, we show that such rescaling is also possible for models with non-local reaction terms, as selection-mutation models. However, to obtain a more relevant qualitative behavior for this second case, we introduce, in the second part of the paper, a second rescaling where we assume that the diffusion steps are small. In this way, using a WKB ansatz, we 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. However, the rescaling introduced here is very different from the latter cases. We extend these results to the multidimensional case.