  In this paper we analyze the decay and the growth for large time of weak and
strong solutions to the three-dimensional viscous Boussinesq system. We show
that generic solutions blow up as $t\to\infty$ in the sense that the energy and
the $L^p$-norms of the velocity field grow to infinity for large time, for
$1\le p<3$. In the case of strong solutions we provide sharp estimates both
from above and from below and explicit asymptotic profiles. We also show that
solutions arising from $(u_0,\theta_0)$ with zero-mean for the initial
temperature $\theta_0$ have a special behavior as $|x|$ or $t$ tends to
infinity: contrarily to the generic case, their energy dissipates to zero for
large time.
