  We study the open quantum dynamics of the Sachdev-Ye-Kitaev (SYK) model
described by the Lindblad master equation, where the SYK model is coupled to
Markovian reservoirs with jump operators that are either linear or quadratic in
the Majorana fermion operators. Of particular interest for us is the time
evolution of the dissipative form factor, which quantifies the average overlap
between the initial and time-evolved density matrices as an open quantum
generalization of the Loschmidt echo. We find that the dissipative form factor
exhibits dynamical quantum phase transitions. We analytically demonstrate a
discontinuous dynamical phase transition in the limit of large number of
fermion flavors, which is formally akin to the thermal phase transition in the
two-coupled SYK model between the black-hole and wormhole phases. We also find
continuous dynamical phase transitions that do not have counterparts in the
two-coupled SYK model. While the phase transitions are sharp in the limit of
large number of fermion flavors, their qualitative signatures are present even
for the finite number of fermion flavors, as we show numerically.
