Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator

Abstract

Abstract In this article, we consider the parabolic equations with nonlocal Monge-Ampère operators ∂ u ∂ t ( x , t ) − D s θ u ( x , t ) = f ( u ( x , t ) ) , ( x , t ) ∈ R + n × R . \frac{\partial u}{\partial t}\left(x,t)-{D}_{s}^{\theta }u\left(x,t)=f\left(u\left(x,t)),\hspace{1.0em}\left(x,t)\in {{\mathbb{R}}}_{+}^{n}\times {\mathbb{R}}. We first prove the narrow region principle and maximal principle for antisymmetric functions, under the condition that u u is uniformly bounded, which weaken the general decay condition u → 0 u\to 0 at infinity. Then, the monotonicity of positive solutions is established using the method of moving planes.