  In this paper we study the $L^2$-gradient flow of the penalized elastic
energy on networks of $q$-curves in $\R^{n}$ for $q \geq 3$. Each curve is
fixed at one end-point and at the other is joint to the other curves at a
movable $q$-junction. For this geometric evolution problem with natural
boundary condition we show the existence of smooth solutions for a (possibly)
short interval of time. Since the geometric problem is not well-posed, due to
the freedom in reparametrization of curves, we consider a fourth-order
non-degenerate parabolic quasilinear system, called the analytic problem, and
show first a short-time existence result for this parabolic system. The proof
relies on applying Solonnikov's theory on linear parabolic systems and Banach
fixed point theorem in proper H\"{o}lder spaces. Then the original geometric
problem is solved by establishing the relation between the analytical solutions
and the solutions to the geometrical problem.
