Exponential stability of a traveling wave for an area preserving curvature motion with two endpoints moving freely on a line

The asymptotic behavior of solutions to an area-preserving curvature flow of planar curves in the upper half plane is investigated. Two endpoints of the curve slide along the horizontal axis with prescribed fixed contact angles. First, by establishing an isoperimetric inequality, we prove the global existence of the solution. We then study the asymptotic behavior of solutions with concave initial data near a traveling wave.