Monotone Positive Radial Solution of Double Index Logarithm Parabolic Equations

This article mainly studies the double index logarithmic nonlinear fractional g-Laplacian parabolic equations with the Marchaud fractional time derivatives ∂tα. 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 g-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 g-Laplacian parabolic equations is studied.