In this paper, we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. We first obtain a series of needed key ingredients such as narrow region principles, and various asymptotic maximum and strong maximum principles for antisymmetric functions in both bounded and unbounded domains. Then we illustrate how this new method can be employed to obtain asymptotic radial symmetry and monotonicity of positive solutions in a unit ball and on the whole space. Namely, we show that no matter what the initial data are, the solutions will eventually approach to radially symmetric functions. We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal parabolic problems with more general operators and more general nonlinearities.

中文翻译：

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分数式抛物线方程移动平面的渐近方法

在本文中，我们开发了一种应用移动平面的渐近方法来研究分数抛物线方程正解的定性性质的系统方法。我们首先获得了一系列需要的关键成分，例如窄域原理，以及有界和无界域中反对称函数的各种渐近最大值和强最大值原理。然后我们说明了如何使用这种新方法来获得单位球和整个空间上的渐近径向对称性和正解的单调性。也就是说，我们表明无论初始数据是什么，解最终都会接近径向对称函数。