Large Time Behavior of Exponential Surface Diffusion Flows on R

Abstract

We consider a surface diffusion flow of the form $V={\mathrm{\partial }}_s^{2}f(-\kappa )$ with a strictly increasing smooth function $f$ , typically $f(r)=e^r$ , for curves, where $s$ denotes the arc-length parameter, $\kappa$ denotes the curvature, and $V$ denotes the normal velocity. The conventional surface diffusion flow corresponds to the case when $f(r)=r$ . We consider this equation for graphs of time-dependent functions defined on the whole real line $\mathbb{R}$ . We prove that there exists a unique global-in-time classical solution with the initial data $u_0$ , provided that the first and the second derivatives of $u_0$ are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation $V=-f^{\prime }(0)\kappa$ . Our result justifies Mullins' grooving model directly obtained by Gibbs–Thomson law without linearization of $f$ near $\kappa =0$ .