Large time decay properties of solutions to a viscous Boussinesq system in a half space

We consider the long time behavior of weak and strong solutions of the n-dimensional viscous Boussinesq system in the half space, with $n\ge 3$ . The $L^r(R_{+}^n)$-asymptotics of strong solutions and their first three derivatives, with $1\le r\le \mathrm{\infty }$, are derived combining $L^q-L^r$ estimates and properties of the fractional powers of the Stokes operator. For the $L^{\mathrm{\infty }}-$asymptotics of the second order derivatives the unboundedness of the projection operator $P:L^{\mathrm{\infty }}(R_{+}^n)\to L_{\sigma }^{\mathrm{\infty }}(R_{+}^n)$ is dealt by an appropriate decomposition of the nonlinear term.