  We carry out a systematical study of the size scaling of Liouvillian gap in
boundary-dissipated one-dimensional quasiperiodic and disorder systems. By
treating the boundary-dissipation operators as a perturbation, we derive an
analytical expression of the Liouvillian gap, which indicates clearly the
Liouvillian gap being proportional to the minimum of boundary densities of
eigenstates of the underlying Hamiltonian, and thus give a theoretical
explanation why the Liouvillian gap has different size scaling relation in the
extended and localized phase. While the Liouvillian gap displays a power-law
size scaling $\Delta_{g}\propto L^{- 3}$ in the extended phase, our analytical
result unveils that the Liouvillian gap fulfills an exponential scaling
relation $\Delta_{g}\propto e^{- \kappa L}$ in the localized phase, where
$\kappa$ takes the largest Lyapunov exponent of localized eigenstates of the
underlying Hamiltonian. By scrutinizing the extended Aubry-Andr\'{e}-Harper
model, we numerically confirm that the Liouvillian gap fulfills the exponential
scaling relation and the fitting exponent $\kappa$ coincides pretty well with
the analytical result of Lyapunov exponent. The exponential scaling relation is
further verified numerically in other one-dimensional quasiperiodic and random
disorder models. We also study the relaxation dynamics and show the inverse of
Liouvillian gap giving a reasonable timescale of asymptotic convergence to the
steady state.
