Global existence and convergence of $Q$-curvature flow on manifolds of even dimension

Using a negative gradient flow approach, we generalize and unify some existence theorems for the problem of prescribing $Q$-curvature first by Baird, Fardoun, and Regbaoui (Calc. Var. 27 75-104) for $4$-manifolds with a possible sign-changing curvature candidate then by Brendle (Ann. Math. 158 323-343) for $n$-manifolds with even $n$ with positive curvature candidate to the case of $n$-manifolds of all even dimension with sign-changing curvature candidates. Making use of the L ojasiewicz--Simon inequality, we also address the rate of the convergence.