Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem

In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic non-homogeneous curvature flows without global forcing terms. By the stationary solutions of such anisotropic flows, we obtain existence results for a class of dual Orlicz Christoffel-Minkowski type problems, which is equivalent to solve the PDE $G(x,u_K,D{u}_K)F(D^2{u}_K+u_{K}I)=1$ on ${\mathbb{S}}^n$ for a convex body $K$, where $D$ is the covariant derivative with respect to the standard metric on ${\mathbb{S}}^n$ and $I$ is the unit matrix of order $n$. This result covers many previous known solutions to $L^p$ dual Minkowski problem, $L^p$ dual Christoffel-Minkowski problem, and some dual Orlicz Minkowski problem etc… Meanwhile, the variational formula of some modified quermassintegrals and the corresponding prescribed area measure problem (Orlicz Christoffel-Minkowski type problem) are considered, and inequalities involving modified quermassintegrals are also derived. As corollary, this gives a partial answer about the general prescribed curvature problem raised in Guan-Ren-Wang (CPAM, 2015).