Monotonic Behavior of Positive Solutions for Semi-Linear Parabolic Equations with Uniformly Elliptic Non-local Operators in Half-Space

Abstract

We address the problem given by the following partial differential equation: some semi-Linear parabolic equations with uniformly elliptic non-local operators in Half-Space. Initially, we establish a generalized weighted average inequality and a maximum principle in unbounded domains, which are crucial for the sliding method. Then, we employ sliding to demonstrate the monotonicity of bounded positive solutions. In this paper, we will remove the monotonicity assumption of the kernel function $a(x)$ by using the sliding method. The techniques employed in the process of this method have applications to other problems related to uniformly elliptic operators.