  On a closed Riemannian manifold $(M,g_0)$ of even dimension $n \geqslant 4$,
the well-known prescribed $Q$-curvature problem asks whether or not there is a
metric $g$ comformal to $g_0$ such that its $Q$-curvature, associated with the
GJMS operator $\mathbf P_g$, is equal to a given function $f$. Letting $g =
e^{2u}g_0$, this problem is equivalent to solving \[ \mathbf P_{g_0} u+Q_{g_0}
= f e^{nu}, \] where $Q_{g_0}$ denotes the $Q$-curvature of $g_0$. The primary
objective of the paper is to introduce the following negative gradient flow of
the time dependent metric $g(t)$ conformal to $g_0$, \[ \frac{\partial g
(t)}{\partial t}= -2\Big(Q_{g (t)} - \frac{\int_M f Q_{g(t)} d\mu_{g(t)}
}{\int_M f^2 d\mu_{g(t)} }f \Big)g(t) \quad \text{ for } t >0, \] to study the
problem of prescribing $Q$-curvature. Since $\int_M Q_g d\mu_g$ is conformally
invariant, our analysis depends on the size of $\int_M Q_{g_0} d\mu_{g_0}$,
which is assumed to satisfy \[ \int_M Q_0 d\mu_{g_0} \ne k (n-1)! \, {\rm
vol}(\mathbb S^n) \quad \text{ for all } \; k = 2,3,... \] The paper is
twofold. First, we identify suitable conditions on $f$ such that the gradient
flow defined as above is defined to all time and convergent, as time goes to
infinity, sequentially or uniformly. Second, we show that various existence
theorems for prescribed $Q$-curvature problem can be derived from the
convergence of the flow.
