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 → ∞ t in the sense that the energy and the L p Lᵖ -norms of the velocity field grow to infinity for large time for 1 ≤ p > 3 1 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, θ 0) (u₀, ₀) with zero-mean for the initial temperature θ 0 ₀ have a special behavior as | x | |x| or t t tends to infinity: contrary to the generic case, their energy dissipates to zero for large time.