A Liouville theorem and radial symmetry for dual fractional parabolic equations.

In this paper, we first study the dual fractional parabolic equation ∂ t α u (x , t) + (− Δ) s u (x , t) = f (u (x , t)) in  B 1 (0) × ℝ , subjected to the vanishing exterior condition. We show that for each t ∈ ℝ , the positive bounded solution u (⋅ , t) must be radially symmetric and strictly decreasing about the origin in the unit ball in ℝ n . To overcome the challenges caused by the dual nonlocality of the operator ∂ t α + (− Δ) s , some novel techniques were introduced. Then we establish the Liouville theorem for the homogeneous equation in the whole space ∂ t α u (x , t) + (− Δ) s u (x , t) = 0 in  ℝ n × 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 ∂ t α u (t) = 0 in  ℝ. To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator ∂ t α , 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 ∂ t α , including even more general Marchaud time derivatives. [ABSTRACT FROM AUTHOR]