  We study the quasinormal modes and non-linear dynamics of a simplified model
of semi-holography, which consistently integrates mutually interacting
perturbative and strongly coupled holographic degrees of freedom such that the
full system has a total conserved energy. We show that the thermalization of
the full system can be parametrically slow when the mutual coupling is weak.
For typical homogeneous initial states, we find that initially energy is
transferred from the black brane to the perturbative sector, later giving way
to complete transfer of energy to the black brane at a slow and constant rate,
while the entropy grows monotonically for all time. Larger mutual coupling
between the two sectors leads to larger extraction of energy from the black
brane by the boundary perturbative system, but also quicker irreversible
transfer of energy back to the black brane. The quasinormal modes replicate
features of a dissipative system with a softly broken symmetry including the
so-called k-gap. Furthermore, when the mutual coupling is below a critical
value, there exists a hybrid zero mode with finite momentum which becomes
unstable at higher values of momentum, indicating a Gregory-Laflamme type
instability. This could imply turbulent equipartitioning of energy between the
boundary and the holographic degrees of freedom in the presence of
inhomogeneities.
