Asymptotic method of moving planes for fractional parabolic equations

TL;DR

This paper introduces an asymptotic method of moving planes to analyze positive solutions of fractional parabolic equations, establishing their eventual radial symmetry and monotonicity regardless of initial data.

Contribution

The paper develops a systematic approach with new principles for fractional parabolic equations, enabling analysis of asymptotic symmetry and monotonicity of solutions.

Findings

Solutions become radially symmetric over time.

Established maximum principles for antisymmetric functions.

Method applicable to various nonlocal parabolic problems.

Abstract

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.