  We study the regimes of heating in the periodically driven $O(N)$-model,
which represents a generic model for interacting quantum many-body systems. By
computing the absorbed energy with a non-equilibrium Keldysh Green's function
approach, we establish three dynamical regimes: at short times a
single-particle dominated regime, at intermediate times a stable Floquet
prethermal regime in which the system ceases to absorb, and at parametrically
late times a thermalizing regime. Our simulations suggest that in the
thermalizing regime the absorbed energy grows algebraically in time with an the
exponent that approaches the universal value of $1/2$, and is thus
significantly slower than linear Joule heating. Our results demonstrate the
parametric stability of prethermal states in a generic many-body system driven
at frequencies that are comparable to its microscopic scales. This paves the
way for realizing exotic quantum phases, such as time crystals or interacting
topological phases, in the prethermal regime of interacting Floquet systems.
