  In this paper, we study the super-critical Quasi-Geostrophic equation in
Gevrey-Sobolev space. We prove the local existence of $(QG)$ for any large
initial data and we give an exponential type of Blow-up to the solution.
Moreover, we establish the existence global for a small initial data and we
show that $\|\theta\|_{H^s_{a, \alpha^{-1}}}$ decays to zero as time goes to
infinity. Fourier analysis and standard techniques are used.
