标题： Fisher-KPP方程中混合算子的传播 显示英文标题

标题： Propagation in the Fisher-KPP equation with Mixed Operator

摘要： 我们的研究聚焦于带有混合局部-非局部算子的Fisher-KPP方程在扩散中的渐近传播行为（参见X. Cabré和J.-M. Roquejoffre于2013年的研究，参考文献[8]）到混合扩散设置中的情况，该设置同时包含经典拉普拉斯算子和分数阶拉普拉斯算子，以分析方程的长时间动力学。我们方法中的一个关键步骤涉及与混合算子相关的热核的构建和详细研究，我们利用此来发展温和解的理论，并在合适的加权函数空间中建立比较原理。这个框架使我们能够严格证明行波的不存在性，并表征解的大时间传播速率。我们表明分数阶拉普拉斯算子的影响在经典拉普拉斯算子之上，特别是在初始层中，它决定了解尾部的指数传播速率和厚度。 显示英文摘要

摘要： Our investigation focuses on the asymptotic spreading behavior of the Fisher-KPP equation with a mixed local-nonlocal operator in the diffusion (see the work by X. Cabr\'e and J.-M. Roquejoffre, 2013, ref.[8]) to the setting of mixed diffusion, which involves both the classical and the fractional Laplacian in order to analyze the long-time dynamics of the equation. A key step in our approach involves the construction and detailed study of the heat kernel associated with the mixed operator, which we use to develop a theory of mild solutions and establish a comparison principle in suitable weighted function spaces. This framework allows us to rigorously establish the non-existence of traveling waves and characterize the large-time spreading rate of solutions. We show that the influence of the fractional Laplacian dominates over the classical Laplacian, especially in the initial layer, where it dictates the exponential propagation rate and the thickness of the solution tails.