  We prove existence and uniqueness of the motion by curvatureof networks in
$\mathbb{R}^n$ when the initial datum is of class $W^{2-\frac{2}{p}}_p$, with
triple junction where the unit tangent vectors to the concurring curves form
angles of $120$ degrees. Moreover we investigated the regularization effect due
to the parabolic nature of the system. An application of this wellposedness
result is a new proof of Theorem 3.18 in "Motion by Curvature of Planar
Networks" by Mantegazza-Novaga-Tortorelli where the possible behaviors of the
solutions at the maximal time of existence are described. Our study is
motivated by an open question proposed in "Evolution of Networks with Multiple
Junctions " by Mantegazza-Novaga-Pluda-Schulze: does there exist a unique
solution of the motion by curvature of networks with initial datum a regular
network of class $C^2$? We give a positive answer.
