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Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), weconsider strictly mean convex hypersurfaces with boundary which are star-shapedwith respect to the center of the cone and which meet the cone perpendicularly.If those hypersurfaces inside the cone evolve by a class of inverse curvatureflows, then, by using the convexity of the cone in the derivation of thegradient and H\"{o}lder estimates, we can prove that this evolution exists forall the time and the evolving hypersurfaces converge smoothly to a piece of around sphere as time tends to infinity.

A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone