Q-curvature flow on S^4

We study a natural counterpart of the Nirenberg problem, namely to prescribe the Q-curvature of a conformal metric on the standard S^4 as a given function f. Our approach uses a geometric flow within the conformal class, which either leads to a solution of our problem as, in particular, in the case when f ≡ const, or otherwise induces a blow-up of the metric near some point of S4. Under suitable assumptions on f, also in the latter case the asymptotic behavior of the flow gives rise to existence results via Morse theory