  We consider the long time behavior of weak and strong solutions of the
n-dimensional viscous Boussinesq system in the half space, with $n\geq3$ . The
$L^r(R^n_+)$-asymptotics of strong solutions and their first three derivatives,
with $1\leq r\leq\infty$, are derived combining $L^q-L^r$ estimates and
properties of the fractional powers of the Stokes operator. For the
$L^\infty-$asymptotics of the second order derivatives the unboundedness of the
projection operator $P: L^\infty(R^n_+)\rightarrow L^\infty_\sigma(R^n_+)$ is
dealt by an appropriate decomposition of the nonlinear term.
