In this talk, I will discuss a curvature-dependent motion of plane curves in a two-dimensional infinite cylinder with spatially undulating boundaries. The two ends of the curve slide freely along the boundary of the domain while keeping the constant contact angle of $\pi/2$.

The question is whether the curve continues to travel to infinity (propagation) or remains in a bounded area (blocking). The same problem was studied in my earlier work under the assumption that the maximum opening angle and closing angle of the boundaries are less than $\pi/4$ (2006 and 2013, joint work with K.-I. Nakamura and B. Lou). Under this condition, one can show that no singularity develops and the solution remains classical. If the opening and closing angles are larger than $\pi/4$, the middle part of the curve may bump into the boundary of the domain, thereby creating singularities. In this talk, I will discuss the long-time behavior of curves that propagates while creating singularities. Among other things we show the existence of such traveling waves and also show that if the bumps are arrayed more densely on the boundary, then the speed of propagation increases. In other words, denser obstacles can facilitate propagation, which may sound somewhat paradoxical. This is joint work with Ryunosuke Mori.