  It is highly nontrivial to what extent we can deduce the relaxation behavior
of a quantum dissipative system from the spectral gap of the Liouvillian that
governs the time evolution of the density matrix. We investigate the relaxation
processes of a quantum dissipative system that exhibits the Liouvillian skin
effect, which means that the eigenmodes of the Liouvillian are localized
exponentially close to the boundary of the system, and find that the timescale
for the system to reach a steady state depends not only on the Liouvillian gap
$\Delta$ but also on the localization length $\xi$ of the eigenmodes. In
particular, we show that the longest relaxation time $\tau$ that is maximized
over initial states and local observables is given by $\tau \sim
\Delta^{-1}(1+L/\xi)$ with $L$ being the system size. This implies that the
longest relaxation time can diverge for $L \to \infty$ without gap closing.
