  We establish a new non-equilibrium scaling regime in the short time evolution
of one-dimensional interacting open quantum systems subject to a generic
heating mechanism. This dynamical regime is characterized by uncompensated
phonon production and a super-diffusive, universal scaling of quasiparticle
lifetimes with momentum $\sim q^{-5/3}$, distinct from finite and zero
temperature cases. It is separated from a high momentum regime by a time
dependent scale fading out as $q_0(t) \sim t^{-4/5}$. In the latter region we
observe thermalization to an effective time-dependent equilibrium with linearly
increasing temperature. By mapping out the dynamical phase diagram and
computing the dynamical structure factor within an open system Keldysh
functional integral approach, we show how these predictions can be explored in
cold atom experiments by means of Bragg spectroscopy.
