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| We study a free boundary problem associated with the curvature
dependent motion of planar curves in the upper half plane whose
two endpoints slide along the horizontal axis with prescribed
fixed contact angles. Our first main result concerns the
classification of solutions; every solution falls into one of
the three categories, namely, area expanding, area bounded and
area shrinking types. We then study in detail the asymptotic
behavior of solutions in each category. Among other things we
show that solutions are asymptotically self-similar both in
the area expanding and the area shriknking cases, while solutions
converge to either a stationary solution or a traveling wave in
the area bounded case. We also prove results on the concavity
properties of solutions. |
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