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It is highly nontrivial to what extent we can deduce the relaxation behaviorof a quantum dissipative system from the spectral gap of the Liouvillian thatgoverns the time evolution of the density matrix. We investigate the relaxationprocesses of a quantum dissipative system that exhibits the Liouvillian skineffect, which means that the eigenmodes of the Liouvillian are localizedexponentially close to the boundary of the system, and find that the timescalefor 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. Inparticular, we show that the longest relaxation time $\tau$ that is maximizedover initial states and local observables is given by $\tau \sim\Delta^{-1}(1+L/\xi)$ with $L$ being the system size. This implies that thelongest relaxation time can diverge for $L \to \infty$ without gap closing.

Liouvillian Skin Effect: Slowing Down of Relaxation Processes without Gap Closing