  In this paper we analyze the motion of a network of three planar curves with
a speed proportional to the curvature of the arcs, having perpendicular
intersections with the outer boundary and a common intersection at a triple
junction. As a main result we show that a linear stability criterion due to
Ikota and Yanagida is also sufficient for nonlinear stability. We also prove
local and global existence of classical smooth solutions as well as various
energy estimates. Finally, we prove exponential stabilization of an evolving
network starting from the vicinity of a linearly stable stationary network.
