  In this paper, we establish a generalized version of Gibbons' conjecture in
the context of the master equation \begin{equation*}
  (\partial_t-\Delta)^s u(x,t)=f(t,u(x,t)) \,\, \mbox{in}\,\,
\mathbb{R}^n\times\mathbb{R}. \end{equation*} We show that, for each
$t\in\mathbb{R}$, the bounded entire solution $u(x,t)$ must be monotone
increasing in one direction, and furthermore it is one-dimensional symmetric
under certain uniform convergence assumption on $u$ and an appropriate
decreasing condition on $f$. These conditions are slightly weaker than their
counter parts proposed in the original Gibbons' conjecture.
  To overcome the difficulties in proving the Gibbons' conjecture and the
impediments caused by the strong correlation between space and time of fully
fractional heat operator $(\partial_t-\Delta)^s$, we introduce some new ideas
and provide several new insights. More precisely, we first derive a weighted
average inequality, which not only provides a straightforward proof for the
maximum principle in bounded domains, but also plays a crucial role in further
deducing the maximum principle in unbounded domains. Such average inequality
and maximum principles are essential ingredients to carry out the sliding
method, and then we apply this direct method to prove the Gibbons' conjecture
in the setting of the master equation.
  It is important to note that the holistic approach developed in this paper is
highly versatile, and will become useful tools in investigating various
qualitative properties of solutions as well as in establishing the Gibbons'
conjecture for a broad range of fractional elliptic and parabolic equations and
systems.
