Asymptotic method of moving planes for fractional parabolic equations
arXiv:2006.14094
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.
arXiv comment40 pages
References in corpus (4)
- Nonlocal diffusion and applications
- Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian
- A maximum principle on unbounded domains and a Liouville theorem for fractional p-harmonic functions
- Maximum principles and monotonicity of solutions for fractional p-equations in unbounded domains