Long-time behavior of damped 2D Boussinesq equations in a half space

We established the global well-posedness and long-time behavior of damped 2D no temperature dissipation Boussinesq equations in half space around the equilibrium state. We first obtained the explicit solution for the linearized system in half space and constructed the relationship between half space and whole space. Then, the resolvent estimates are builded by precious anisotropy analysis for different frequencies. Finally, we got the global solution for the weak dissipation system in half infinite domain and does not required the assistance of decay rate.