  We study fractional parabolic equations with indefinite nonlinearities $$
\frac{\partial u} {\partial t}(x,t) +(-\Delta)^s u(x,t)= x_1 u^p(x, t),\,\, (x,
t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We
first prove that all positive bounded solutions are monotone increasing along
the $x_1$ direction. Based on this we derive a contradiction and hence obtain
non-existence of solutions. These monotonicity and nonexistence results are
crucial tools in a priori estimates and complete blow-up for fractional
parabolic equations in bounded domains. To this end, we introduce several new
ideas and developed a systematic approach which may also be applied to
investigate qualitative properties of solutions for many other fractional
parabolic problems.
