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In this paper, we study the subcritical dissipative quasi-geostrophic equation. By using the Littlewood Paley theory, Fourier analysis and standard techniques we prove that there exists v a unique global-in-time solution for small initial data belonging to the critical Fourier-Besov-Morrey spaces F N _p, λ, q^3-2 α+λ-2p . Moreover, we show the asymptotic behavior of the global solution v. i.e., \|v(t)\|_ F N _p, λ, q^3-2 α+λ-2p decays to zero as time goes to infinity. , number=1, journal=Journal of Partial Differential Equations, year=2023, month=Jun., pages=1–21