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

Abstract

In this article, we consider the parabolic equations with nonlocal Monge-Ampère operators ∂u∂t(x,t)−Dsθu(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 uu is uniformly bounded, which weaken the general decay condition u→0u\to 0 at infinity. Then, the monotonicity of positive solutions is established using the method of moving planes.

Keywords

• parabolic equations
• nonlocal monge-ampère operator
• method of moving planes
• monotonicity
• 35r11
• 35k58