Summary: In this paper, we first study the dual fractional parabolic equation \[ \partial_t^{\alpha}u(x,t)+ (-\Delta)^s u(x,t)=f(u(x,t))\quad\text{in } B_1 (0)\times\mathbb{R}, \] subjected to the vanishing exterior condition. We show that for each \(t\in\mathbb{R}\), the positive bounded solution \(u(\cdot,t)\) must be radially symmetric and strictly decreasing about the origin in the unit ball in \(\mathbb{R}^n\). To overcome the challenges caused by the dual nonlocality of the operator \(\partial_t^{\alpha}+ (-\Delta)^s\), some novel techniques were introduced.
Then we establish the Liouville theorem for the homogeneous equation in the whole space \[ \partial_t^{\alpha}u(x,t)+(-\Delta)^s u(x,t)=0\quad\text{in } \mathbb{R}^n \times\mathbb{R}. \] We first prove a maximum principle in unbounded domains for antisymmetric functions to deduce that \(u(x,t)\) must be constant with respect to \(x\). Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation \[ \partial_t^{\alpha}u(t)=0 \quad\text{in }\mathbb{R}. \] To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator \(\partial_t^{\alpha}\), we bring in some new ideas and simpler approaches. Instead of disturbing the antisymmetric function, we employ a perturbation technique directly on the solution \(u(t)\) itself. This method provides a more concise and intuitive way to establish the Liouville theorem for one-sided operators \(\partial_t^{\alpha}\), including even more general Marchaud time derivatives.