Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2021; 58(4): 1001-1017

Published online July 1, 2021 https://doi.org/10.4134/JKMS.j200418

Copyright © The Korean Mathematical Society.

Symmetry and monotonicity of solutions to fractional elliptic and parabolic equations

Fanqi Zeng

Xinyang Normal University

Abstract

In this paper, we first apply parabolic inequalities and a maximum principle to give a new proof for symmetry and monotonicity of solutions to fractional elliptic equations with gradient term by the method of moving planes. Under the condition of suitable initial value, by maximum principles for the fractional parabolic equations, we obtain symmetry and monotonicity of positive solutions for each finite time to nonlinear fractional parabolic equations in a bounded domain and the whole space. More generally, if bounded domain is a ball, then we show that the solution is radially symmetric and monotone decreasing about the origin for each finite time. We firmly believe that parabolic inequalities and a maximum principle introduced here can be conveniently applied to study a variety of nonlocal elliptic and parabolic problems with more general operators and more general nonlinearities.

Keywords: The fractional parabolic equation, monotonicity, symmetry, the method of moving planes, fractional elliptic equations with gradient term, maximum principle

MSC numbers: Primary 35K58, 35A09, 35B06, 35B09

Supported by: This work was financially supported by NSFC 11971415, and Henan Province Science Foundation for Youths (No.212300410235), and the Key Scientific Research Program in Universities of Henan Province (No.21A110021), and Nanhu Scholars Program for Young Scholars of XYNU (No. 2019).