Dimension-Dependent Asymptotic Dynamics for Mean Curvature Flow with Robin Boundary Conditions

Abstract

We consider a graphical mean curvature flow in a cylinder with Robin boundary conditions, which arises as a geometric model for interface motion in the singular limit of the Allen--Cahn equation with nonlinear boundary conditions. It was shown in [26] that, in the planar case, every solution converges to a translating Grim Reaper with a \emph{fixed profile} and \emph{finite speed}. In this paper, we investigate the radially symmetric problem in higher dimensions and reveal a completely different asymptotic dynamics caused by the spatial dimension. In contrast to the planar case, there is no fixed translating profile governing the long-time behaviour. Instead, the solution propagates with an exponentially increasing speed, while both the gradient $|Du|$ (away from the center) and the instantaneous speed $u_t$ diverge exponentially as $t\to\infty$. This reveals a fundamentally different asymptotic behaviour induced by the interaction between the Robin boundary condition and the spatial dimension, that is, the translating profile continuously degenerates and becomes asymptotically ray-like. Since the equation becomes asymptotically degenerate and no uniform-in-time $C^0$, $C^1$, or $C^2$ estimates are available, our analysis relies on a new approach based on the zero number argument.