  In this paper, we consider the following indefinite fully fractional heat
equation involving the master operator \begin{equation*} (\partial_t
-\Delta)^{s} u(x,t) = x_1u^p(x,t)\ \ \mbox{in}\ \R^n\times\R , \end{equation*}
where $s\in(0,1)$, and $-\infty < p < \infty$. Under mild conditions, we prove
that there is no positive bounded solutions. To this end, we first show that
the solutions are strictly increasing along $x_1$ direction by employing the
direct method of moving planes. Then by constructing an unbounded sub-solution,
we derive the nonexistence of bounded solutions.
  To circumvent the difficulties caused by the fully fractional master
operator, we introduced some new ideas and novel approaches that, as we
believe, will become useful tool in studying a variety of other fractional
elliptic and parabolic problems.
