  Given a smooth convex cone in the Euclidean $(n+1)$-space ($n\geq2$), we
consider strictly mean convex hypersurfaces with boundary which are star-shaped
with 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 curvature
flows, then, by using the convexity of the cone in the derivation of the
gradient and H\"{o}lder estimates, we can prove that this evolution exists for
all the time and the evolving hypersurfaces converge smoothly to a piece of a
round sphere as time tends to infinity.
