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This paper concerns the evolution of complete noncompact locally uniformlyconvex hypersurface in Euclidean space by curvature flow, for which the normalspeed $\Phi$ is given by a power $\beta\geq 1$ of a monotone symmetric andhomogeneous of degree one function $F$ of the principal curvatures. Under theassumption that $F$ is inverse concave and its dual function approaches zero onthe boundary of positive cone, we prove that the complete smooth strictlyconvex solution exists and remains a graph until the maximal time of existence.In particular, if $F=K^{s/n}G^{1-s}$ for any $s\in(0, 1]$, where $G$ is ahomogeneous of degree one, increasing in each argument and inverse concavecurvature function, we prove that the complete noncompact smooth strictlyconvex solution exists and remains a graph for all times.

Evolution of complete noncompact graphs by powers of curvature function