  In this paper, we first address the space-time decay properties for higher
order derivatives of strong solutions to the Boussinesq system in the usual
Sobolev space. The decay rates obtained here are optimal. The proof is based on
a parabolic interpolation inequality, bootstrap argument and some weighted
estimates. Secondly, we present a new solution integration formula for the
Boussinesq system, which will be employed to establish the existence of strong
solutions in scaling invariant function spaces. We further investigate the
asymptotic profiles and decay properties of these strong solutions. Our results
recover and extend the important results in Brandolese and Schonbek (Tran. A.
M.S. Vol 364, No.10, 2012, 5057-5090).
