  We study the nonlinear fractional reaction diffusion equation $\partial_{t}u
+ (-\Delta)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together
with Dirichlet boundary conditions on $\R^N \setminus \Omega$. We prove
asymptotic symmetry of nonnegative globally bounded solutions in the case where
the underlying data obeys some symmetry and monotonicity assumptions. More
precisely, we assume that $\Omega$ is symmetric with respect to reflection at a
hyperplane, say ${x_1=0}$, and convex in the $x_1$-direction, and that the
nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather
weak additional technical assumptions, we then show that any nonzero element in
the $\omega$-limit set of nonnegative globally bounded solution is even in
$x_1$ and strictly decreasing in $|x_1|$. This result, which is obtained via a
series of new estimates for antisymmetric supersolutions of a corresponding
family of linear equations, implies a strong maximum type principle which is
not available in the non-fractional case $s=1$.
