Liouville type theorems for dual nonlocal evolution equations involving Marchaud derivatives

In this paper, we establish a Liouville type theorem for the homogeneous dual fractional parabolic equation equation ^ₜ u (x, t) + (-) ˢ u (x, t) = 0\ \ in\ \ Rⁿ. equation where 0<, s<1. Under an asymptotic assumption |ₗ|u (x, t) |x|^ 0 \; (or \; 0) \, \, for some \;0 1, in the case 12<s < 1, we prove that all solutions in the sense of distributions of above equation must be constant by employing a method of Fourier analysis. Our result includes the previous Liouville theorems on harmonic functions ABR and on s-harmonic functions CDL as special cases and it is still novel even restricted to one-sided Marchaud fractional equations, and our methods can be applied to a variety of dual nonlocal parabolic problems. In the process of deriving our main result, through very delicate calculations, we obtain an optimal estimate on the decay rate of ₑ₈₆₇ₓ^+ (-) ˢ (x, t) for functions in Schwartz space. This sharp estimate plays a crucial role in defining the solution in the sense of distributions and will become a useful tool in the analysis of this family of equations.