The $$Q_k$$ flow on complete non-compact graphs

We establish the long time existence of complete non-compact weakly convex and smooth hypersurfaces $$\Sigma _t$$ evolving by $$Q_k$$ -flow. show that maximum T depends on dimension $$d_W$$ vector space $$W{:}{=}\{w \in \mathbb {R}^{n+1}: \sup _{X\in \Sigma _0} |\langle X,w\rangle | = +\infty \}$$ which contains each direction in our initial data _0$$ is infinite. If $$d_W=\text {dim}(W) \ge n-k+1$$ , then solution exists for all $$t (0,+\infty )$$ ; if \le n-k$$ exsist up to some finite $$T < $$ . In latter case, trace at infinity $$\Gamma a closed viscosity $$(n-d_W)$$ -dimensional flow (0,T)$$