Large time behavior of exponential surface diffusion flows on $\mathbb{R}$

We consider a surface diffusion flow of the form $V=\partial_s^2f(-\kappa)$ with a strictly increasing smooth function $f$ typically, $f(r)=e^r$, for a curve with arc-length parameter $s$, where $\kappa$ 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 the graph of a function defined on the whole real line $\mathbb{R}$. We prove that there exists a unique global-in-time classical solution provided that the first and the second derivatives are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation $V=-f'(0)\kappa$. Our result justifies the explanation for grooving modeled by Mullins (1957) directly obtained by Gibbs--Thomson law without linearization of $f$ near $\kappa=0$.