Sliding method for equations with uniformly parabolic fractional operators 

Abstract

In this paper, we obtain the one-dimensional symmetry and monotonicity of entire solutions for equations involving uniformly parabolic fractional operators via a sliding method. We first establish a generalized weighted average inequality and the maximum principle in unbounded domains, then using the sliding method, we derive the symmetry and monotonicity of entire solutions to fractional uniformly parabolic equations in the whole space. MSC 2020: 35B50, 35R11, 35K60.