We study the dynamics and trapping of excitations for a dimer with an energy off-set Δ coupled to an external environment. Using a Lindblad quantum master equation approach, we calculate the survival probability Π(t) of the excitation and define different lifetimes τ(s) of the excitation, corresponding to the duration of the decay of Π(t) in between two predefined values. We show that it is not possible to always enhance the overall decay to the trap. However, it is possible, even for not too small environmental couplings and for values of Δ of the order O(1), to decrease certain lifetimes τ(s), leading to faster decay of Π(t) in these time intervals: there is an optimal environmental coupling, leading to a maximal decay for fixed Δ.