| Summary: | This article mainly studies the double index logarithmic nonlinear fractional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mtext>-</mtext></mrow></semantics></math></inline-formula>Laplacian parabolic equations with the Marchaud fractional time derivatives <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mstyle displaystyle="true"><msubsup><mo>∂</mo><mi>t</mi><mi>α</mi></msubsup></mstyle></semantics></math></inline-formula>. Compared with the classical direct moving plane method, in order to overcome the challenges posed by the double non-locality of space-time and the nonlinearity of the fractional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mtext>-</mtext></mrow></semantics></math></inline-formula>Laplacian, we establish the unbounded narrow domain principle, which provides a starting point for the moving plane method. Meanwhile, for the purpose of eliminating the assumptions of boundedness on the solutions, the averaging effects of a non-local operator are established; then, these averaging effects are applied twice to ensure that the plane can be continuously moved toward infinity. Based on the above, the monotonicity of a positive solution for the above fractional <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mtext>-</mtext></mrow></semantics></math></inline-formula>Laplacian parabolic equations is studied.
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