$Q$-curvature flow on closed manifolds of even dimension.

The primary objective of the paper is to use a negative gradient flow method to study the problem of prescribing $Q$-curvature on even dimensional manifolds with sign-changing curvature candidate $f$. Under some proper assumptions on $f$, we are able to prove that the flow exists globally and converges sequentially to a conformal metric. This limit metric has $lambda_infty f$ as its $Q$-curvature for some suitable constant $lambda_infty$. As byproducts, various existence theorems for prescribed $Q$-curvature problem can be derived.