  We theoretically study the dynamical Casimir effect (DCE), i.e., parametric
amplification of a quantum vacuum, in an optomechanical cavity interacting with
a photonic crystal, which is considered to be an ideal system to study the
microscopic dissipation effect on the DCE. Starting from a total Hamiltonian
including the photonic band system as well as the optomechanical cavity, we
have derived an effective Floquet-Liouvillian by applying the Floquet method
and Brillouin-Wigner-Feshbach projection method. The microscopic dissipation
effect is rigorously taken into account in terms of the energy-dependent
self-energy. The obtained effective Floquet-Liouvillian exhibits the two
competing instabilities, i.e., parametric and resonance instabilities, which
determine the stationary mode as a result of the balance between them in the
dissipative DCE. Solving the complex eigenvalue problem of the
Floquet-Liouvillian, we have determined the stationary mode with vanishing
values of the imaginary parts of the eigenvalues. We find a new non-local
multimode DCE represented by a multimode Bogoliubov transformation of the
cavity mode and the photon band. We show the practical advantage for the
observation of DCE in that we can largely reduce the pump frequency when the
cavity system is embedded in a narrow band photonic crystal with a bandgap.
