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

TL;DR

This paper investigates high-dimensional mean curvature flow with Robin boundary conditions, revealing exponential blow-up behavior in the interface speed and gradient, contrasting with finite-speed translation in lower dimensions.

Contribution

It demonstrates that in higher dimensions, the mean curvature flow exhibits exponential growth in speed and gradient, a novel phenomenon not observed in planar cases, using a new zero number argument approach.

Findings

Radial flow propagates at exponential asymptotic speed.

Gradient and instantaneous speed increase exponentially over time.

The equation becomes asymptotically degenerate without uniform estimates.

Abstract

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.