Qualitative properties of solutions for dual fractional parabolic equations involving nonlocal Monge-Ampère operator

Abstract In this article, we mainly study the qualitative properties of solutions for dual fractional-order parabolic equations with nonlocal Monge-Ampère operators in different domains <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msubsup> <m:mrow> <m:mo>∂</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mrow> <m:mi>β</m:mi> </m:mrow> </m:msubsup> <m:mi>μ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>−</m:mo> <m:msubsup> <m:mrow> <m:mi>D</m:mi> </m:mrow> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mrow> <m:mi>τ</m:mi> </m:mrow> </m:msubsup> <m:mi>μ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>μ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>.</m:mo> </m:math> {\partial }_{t}^{\beta }\mu \left(y,t)-{D}_{\alpha }^{\tau }\mu \left(y,t)=f\left(\mu \left(y,t)). We first establish a series of maximum principles and averaging effects theorems for antisymmetric functions and then used the method of moving planes and sliding planes to establish radial symmetry, monotonicity, nonexistence, and Liouville theorem for positive solutions.