Large time decay and growth for solutions of a viscous Boussinesq system

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 $t\to \mathrm{\infty }$ 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 $\mathrm{\mid }x\mathrm{\mid }$ or $t$ tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.