  In this paper, we consider the dual fractional parabolic problem in the right
half space. We prove that the positive solutions are strictly increasing in
$x_1$ direction without assuming the solutions be bounded.
  So far as we know, this is the first paper to explore the monotonicity of
possibly unbounded solutions for the nonlocal parabolic problem involving both
the fractional time derivative $\partial_t^\alpha$ and the fractional Laplacian
$(-\Delta)^s$. To overcome the difficulties caused by the dual nonlocality in
space-time and by the remarkably weak assumptions on solutions, we introduced
several new ideas and our approaches are quite different from those in the
previous literature. We first establish an unbounded narrow region principle
without imposing any decay and boundedness assumptions on the antisymmetric
functions at infinity by estimating the nonlocal operator
$\partial_t^\alpha+(-\Delta)^s$ along a sequence of suitable auxiliary
functions at their minimum points, which is an essential ingredient to carry
out the method of moving planes at the starting point. Then in order to remove
the decay or bounded-ness assumption on the solutions, we develop a new novel
approach lies in establishing the {\em averaging effects} for such nonlocal
operator and apply these {\em averaging effects} twice to guarantee that the
plane can be moved all the way to infinity to derive the monotonicity of
solutions.
  We believe that the new ideas and techniques developed here will become very
useful tools in studying the qualitative properties of solutions, in particular
of those unbounded solutions, for a wide range of fractional elliptic and
parabolic problems.
