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 \begin{equation} \partial^α_t u(x,t)+(-Δ)^s u(x,t) = 0\ \ \mbox{in}\ \ \mathbb{R}^n\times\mathbb{R} . \end{equation} where $0<α,s<1$. Under an asymptotic assumption $$\liminf_{|x|\rightarrow\infty}\frac{u(x,t)}{|x|^γ}\geq 0 \; ( \mbox{or} \; \leq 0) \,\,\mbox{for some} \;0\leqγ\leq 1, $$ in the case $\frac{1}{2}<s < 1$, 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 \cite{ABR} and on $s$-harmonic functions \cite{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 $\left[D_{\rm right}^α+(-Δ)^s\right] φ(x,t)$ 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.

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