On the Type IIb solutions to mean curvature flow

In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph $(y,u(\mathrm{\mid }y\mathrm{\mid }))$ over ${\mathbb{R}}^n$, $n\ge 2$, satisfying there exist positive constants $ϵ$, $c$ and $N$ such that $u^{\mathrm{\prime }}(r)\ge c{r}^{ϵ}$ for $r\ge N$, the longtime solution to mean curvature flow with initial data $(y,u(\mathrm{\mid }y\mathrm{\mid }))$ must be Type IIb. We also study the asymptotic behavior of Type IIb mean curvature flow and show that the limit of suitable rescaling sequence for mean-convex Type IIb mean curvature flow satisfying $\delta$-Andrews’ noncollapsing condition is translating soliton.