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Abstract: Isolated quantum many-body systems which thermalize under their own dynamics are expected to act as their own thermal baths 1, thereby losing memory of initial conditions and bringing their local subsystems to thermal equilibrium. Here 2, we show that the infinite- dimensional limit of a quantum lattice model, as described by dynamical mean-field theory (DMFT), provides a natural framework to understand this self-consistent thermalization process. Using the Fermi-Hubbard model as a working example, we demonstrate that the emergence of a self-consistent bath occurs via a sharp thermalization front, moving ballistically and separating the initial condition from the long time thermal fixed point (Fig. 1). We characterize the full DMFT dynamics through an effective temperature for which we derive a traveling wave equation of the Fisher-Kolmogorov-Petrovsky-Piskunov type 3. We extend our results in order to study the shape of the front and its velocity in open dissipative fermionic systems by integrating DMFT into the Lindblad Master Equation formalism. We show that thermalization under open quantum system dynamics is qualitatively different from the closed-system case. In particular, the thermalization front is strongly modified, a signature of the irreversibility of open-system dynamics 4. 1234R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015) A. Picano, G. Biroli, M. Schiro, Physical Review Letters 134, 116503, (2025). É. Brunet and B. Derrida, J. Stat. Phys. 161, 801 (2015). A. Picano, M. Vanhoecke, M. Schiro, arXiv:2507.21804 (2025).