Radial symmetry, monotonicity and Liouville theorem for Marchaud fractional parabolic equations with the nonlocal Bellman operator

Abstract

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 ${\partial }_{t}^{\alpha }-{F}_{s}$ , 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 ${\partial }_{t}^{\alpha }-{F}_{s}$ . 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.

Keywords

• space-time fractional parabolic equations
• nonlocal bellman operator
• radial symmetry and monotonicity
• liouville theorem
• direct method of moving planes
• 35r11
• 47g30
• 35b50
• 35b53