Radial symmetry, monotonicity and Liouville theorem for Marchaud fractional parabolic equations with the nonlocal Bellman operator - Liu Mengru; Zhang Lihong; Wang Guotao (Advanced Nonlinear Studies)
In this article, we focus on studying space-time fractional parabolic equations with the nonlocal Bellman operator and the Marchaud fractional derivative. To address the difficulty caused by the space-time non-locality of operator ∂tα−Fs ${\partial }_{t}^{α }-F_{s}$ , we first establish the narrow region principle and the maximum principle. Next, we establish a direct method of moving planes applicable to the operator ∂tα−Fs ${\partial }_{t}^{α }-F_{s}$ . Finally, combining perturbation techniques and limit arguments, we apply this direct method of moving planes to prove the radial symmetry and monotonicity of solutions for the space-time fractional parabolic equations involving the nonlocal Bellman operator and the Marchaud fractional derivative. Additionally, the Liouville theorem of homogeneous space-time fractional parabolic equation is proved.