Infinite-Time Blow-up Arising in a Mean Curvature Flow

We consider a mean curvature flow in a cylinder with Robin boundary conditions, which can be used to model the interface motion in singular limit problems of the Allen-Cahn equation with nonlinear boundary conditions. It was shown in \cite{LWY} that the planar curvature flow converges to a translating Grim Reaper with {\it finite speed} and {\it fixed profile}. In this paper we study the high dimensional problem, and show surprisingly different features caused by the dimension: a radial flow $u(|x|,t)$ propagates at {\it exponential asymptotic speed}, both the gradient $|Du|$ (everywhere except for the center) and the instantaneous speed $u_t$ (everywhere) also increase to infinity exponentially as $t\to \infty$. Due to the lack of uniform-in-time $C^0, C^1$ and $C^2$ estimates, the equation is asymptotically degenerate, we will use a new approach (that is, the zero number argument) to prove the conclusions.