PDE Seminar: Matano -- Front propagation through a two-dimensional cylinder with saw-toothed boundaries

PDE Seminar: 11:00am Tuesday 5th December 
Quad S225 

Talk 1.  at 11 am 

Speaker: Hiroshi Matano (Professor @ Meiji University, Japan) 

Title: Front propagation through a two-dimensional cylinder with saw-toothed
boundaries.  

Abstract: 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 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.