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In this paper, we study the long-time existence and asymptotic behavior for aclass of anisotropic inverse Gauss curvature flows. By the stationary solutionsof anisotropic flows, we obtain some new existence results for the dual OrliczMinkowski type problem and even dual Orlicz Minkowski type problem for smoothmeasures, which is the most reasonable extension of the $L^p$ dual Minkowskiproblem from the dual point of view. The results of corresponding $L^p$versions are $L^p$ dual Minkowski problem for $p>q$; and even $L^p$ dualMinkowski problem for $p>-1$, or $q<1$, or some ranges of $p<0<q$, whichcontain all existence results for smooth measures up to now except $p=q$ or$q=n+1$ ($L^p$ Minkowski problem).

A class of anisotropic inverse Gauss curvature flows and dual Orlicz Minkowski type problem