Critical prescribed Q-curvature flow on closed even-dimensional manifolds with sign-changing functions

In this article, we consider the prescribed Q-curvature equation$Pu=\rho (\frac{h{e}^{2nu}}{{\int }_{M}h{e}^{2nu}d\mu }-\frac{1}{|M{|}_{g_0}})\text{in}M,$ where $(M,g_0)$ is a closed 2n-dimensional Riemannian manifold, $P$ represents the GJMS operator, which is (weakly) positive with a kernel of constant functions. The function h is smooth and sign-changing, while ρ is a positive constant. In the critical case with $\rho =4^n(n-1)!{\pi }^n$, we employ a negative gradient-like flow method to establish the existence of solutions to this prescribed Q-curvature equation. Our approach extends the work of Li-Xu [46], which focused on dimension 2, to general even dimensions. This result can also be viewed as a counterpart to [8] in the case where h is a sign-changing function.