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 $0<\alpha ,s<1$. Under an asymptotic assumption $$\underset{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{u(x,t)}{\mathrm{\mid }x{\mathrm{\mid }}^{\gamma }}\ge 0(or\le 0)forsome0\le \gamma \le 1,$$ 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 $[D_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}^{\alpha }+(-\mathrm{\Delta }{)}^s]\phi (x,t)$ 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.