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arXiv · 2609.34504

The fractional Fisher-KPP equation with free boundaries

Huyuan Chen·Yihong Du·Wenjie Ni
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Abstract

In this paper, we consider a free boundary problem with fractional diffusion $(-Δ)^s$ $(0<s<1)$, as a model for species spreading, which can be viewed as a natural extension of the free boundary model in \cite{CDLL2019}, where the nonlocal diffusion term is defined via a continuous integrable kernel function $J(x)$. This fractional diffusion model can also be viewed as a nonlocal version of the free boundary model in \cite{DuLin2010}, whose local diffusion term is given by the classical Laplacian. We first establish the global existence and uniqueness of the solution, which is the main contribution of this paper. This well-posedness question has been open for some time now due to the difficulties caused by the singularity and non-integrability of the associated kernel function of $(-Δ)^s$, and the lack of general enough regularity results for operators extending the standard fractional Laplacian. A crucial step that enables us to answer this question relies on an approximation approach, which breaks an associated linear fractional parabolic initial boundary value problem with curved boundaries into a sequence of approximating problems, where the functions representing the curved boundaries are replaced by approximating step functions. We then show that, for Fisher-KPP type nonlinear growth terms, the long-time dynamics of the model exhibits a spreading-vanishing dichotomy, similar to the earlier models of \cite{DuLin2010, CDLL2019}. More precise descriptions of the spreading profile of the solution will be considered in a separate work.

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Huyuan Chen, Yihong Du, Wenjie Ni. 2026-09-28. The fractional Fisher-KPP equation with free boundaries. https://arxiv.org/abs/2609.34504

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