A Liouville theorem and radial symmetry for dual fractional parabolic equations

Abstract

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