发表机构
Center for Mathematics and Interdisciplinary Sciences, Fudan University; Shanghai Institute for Mathematics and Interdisciplinary Sciences; School of Science and Technology, University of New England（复旦大学数学与交叉科学研究院; 上海数学与交叉科学研究院; 新英格兰大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究分数阶Fisher-KPP方程的自由边界问题，通过近似方法建立解的全局存在性和唯一性，并证明长时间动力学呈现扩展-消失二分法。
AI 中文摘要
在本文中，我们考虑一个具有分数阶扩散项 $(-\Delta)^s$ $(0<s<1)$ 的自由边界问题，作为物种扩散的模型，它可以被视为文献 \cite{CDLL2019} 中自由边界模型的一种自然推广，其中非局部扩散项通过一个连续可积的核函数 $J(x)$ 定义。这个分数阶扩散模型也可以被视为文献 \cite{DuLin2010} 中自由边界模型的非局部版本，其局部扩散项由经典拉普拉斯算子给出。我们首先建立了解的全局存在性和唯一性，这是本文的主要贡献。由于 $(-\Delta)^s$ 的核函数的奇异性和不可积性，以及缺乏足够通用的正则性结果来扩展标准分数阶拉普拉斯算子，这个适定性问题已经悬而未决一段时间。使我们能够回答这个问题的关键步骤依赖于一种近似方法，它将一个相关的具有弯曲边界的线性分数阶抛物型初边值问题分解为一系列近似问题，其中表示弯曲边界的函数被替换为近似的阶梯函数。然后我们证明，对于Fisher-KPP型非线性增长项，模型的长时间动力学表现出一种扩展-消失二分法，类似于文献 \cite{DuLin2010, CDLL2019} 中的早期模型。关于解的扩展轮廓的更精确描述将在另一项工作中考虑。
英文摘要
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.
