Symmetry and monotonicity of solutions to fractional elliptic and parabolic equations
J. Korean Math. Soc. Published online March 31, 2021
School of Mathematics and Statistics, 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 symmetry 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