Asymptotic Monotonicity of Positive Solutions for Fractional Parabolic Equation on the Right Half Space

Abstract

In this paper, we mainly study the asymptotic monotonicity of positive solutions for fractional parabolic equation on the right half space. First, a narrow region principle for antisymmetric functions in unbounded domains is obtained, in which we remarkably weaken the decay condition \(u\rightarrow 0\) at infinity and only assume its growth rate does not exceed \(|x|^{\gamma }\) (\(0 < \gamma < 2s\)) compared with (Adv. Math. 377:107463, 2021). Then we obtain asymptotic monotonicity of positive solutions of fractional parabolic equation on \(\mathbb{R}^{N}_{+}\times (0,\infty )\).