Asymptotic Dynamics of Forced Curvature Flow in a Planar Strip with Robin Boundary Conditions

We study the graphical forced mean curvature flow in a planar strip subject to Robin boundary conditions. The normal velocity is prescribed by $V=H+b$, where $b>0$ is constant and $H$ denotes the curvature of the graph. We consider positive even solutions and reduce the problem to the half interval. The main results describe the degeneration produced by the Robin condition: the height, the instantaneous velocity, and the spatial gradient diverge, while the normalized slope $u_x\mathrm{/}u$ converges to one away from the symmetry axis. The normalized velocity $u_t\mathrm{/}u$ converges to the forcing parameter $b$. The curvature converges to zero in the interior in an integral sense, whereas its horizontal curvature measure concentrates at the symmetry axis.