On a Curvature Flow in a Band Domain with Unbounded Boundary Slopes

We consider an anisotropic curvature flow $V= A(\mathbf{n})H + B(\mathbf{n})$ in a band domain $\Omega :=[-1,1]\times R$, where $\mathbf{n}$, $V$ and $H$ denote the unit normal vector, normal velocity and curvature, respectively, of a graphic curve $\Gamma_t$. We consider the case when $A>0>B$ and the curve $\Gamma_t$ contacts $\partial_\pm \Omega$ with slopes equaling to $\pm 1$ times of its height (which are unbounded when the solution moves to infinity). First, we present the global well-posedness and then, under some symmetric assumptions on $A$ and $B$, we show the uniform interior gradient estimates for the solution. Based on these estimates, we prove that $\Gamma_t$ converges as $t\to \infty$ in $C^{2,1}_{\text{loc}} ((-1,1)\times R)$ topology to a cup-like traveling wave with {\it infinite} derivatives on the boundaries.