  This paper has two primary objectives. The first one is to demonstrate that
the solutions of master equation \begin{equation*} (\partial_t-\Delta)^s u(x,t)
=f(u(x, t)), \,\,(x, t)\in B_1(0)\times \mathbb{R}, \end{equation*} subject to
the vanishing exterior condition, are radially symmetric and strictly
decreasing with respect to the origin in $B_1(0)$ for any $t\in \mathbb{R}$.
Another one is to establish the Liouville theorem for homogeneous master
equation \begin{equation*}
  (\partial_t-\Delta)^s u(x,t)=0 ,\,\, \mbox{in}\,\,
\mathbb{R}^n\times\mathbb{R}, \end{equation*} which states that all bounded
solutions must be constant. We propose a new methodology for a direct method of
moving planes applicable to the fully fractional heat operator
$(\partial_t-\Delta)^s$, and the proof of our main results based on this direct
method involves the perturbation technique, limit argument as well as Fourier
transform. This study opens up a way to investigate the geometric behavior of
master equations, and provides valuable insights for establishing qualitative
properties of solutions and even for deriving important Liouville theorems for
other types of fractional order parabolic equations.
