  We study the relaxation dynamics of interacting, one-dimensional fermions
with band curvature after a weak quench in the interaction parameter. After the
quench, the system is described by a non-equilibrium initial state, which
relaxes towards thermal equilibrium, featuring prethermal behavior on
intermediate time and length scales. The model corresponds to the class of
interacting Luttinger Liquids, which extends the quadratic Luttinger theory by
a weak integrability breaking phonon scattering term. In order to solve for the
non-equilibrium time evolution, we use kinetic equations for the phonon
densities, exploiting the resonant but subleading character of the phonon
interaction term. The interplay between phonon scattering and the quadratic
Luttinger theory leads to the emergence of three distinct spatio-temporal
regimes for the fermionic real-space correlation function. It features the
crossover from a prequench to a prethermal state, finally evolving towards a
thermal state on increasing length and time scales. The characteristic
algebraically decaying real-space correlations in the prethermalized regime
become modulated by an amplitude, which, as an effect of the interactions, is
decaying in time according to a stretched-exponential, while in the thermal
regime exponentially decaying real-space correlations emerge. The asymptotic
thermalization dynamics is governed by energy transport over large distances
from the thermalized to the non-thermalized regions, which is carried out by
macroscopic, dynamical slow modes. This is revealed in an algebraic decay of
the system's effective temperature. The numerical value of the associated
exponent agrees with the dynamical critical exponent of the Kardar-Parisi-Zhang
universality class, which shares with the present interacting Luttinger Liquid
the conservation of total energy and momentum.
