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 systems 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.
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