Asymptotic convergence for modified scalar curvature flow

In this paper, we study the flow of closed, starshaped hypersurfaces in $\mathbb{R}^{n+1}$ with speed $r^\alpha\sigma_2^{1/2},$ where $\sigma_2^{1/2}$ is normalized square root scalar curvature, $\alpha\geq 2,$ and $r$ distance from points on hypersurface to origin. We prove that exists for all time starshapedness preserved. Moreover, after normalization, show converges exponentially fast a sphere centered at When $\alpha<2,$ counterexample given above convergence.