  We study the asymptotic behavior of the trajectory of a nonautonomous
evolution equation governed by a quasi-nonexpansive operator in Hilbert spaces.
We prove the weak convergence of the trajectory to a fixed point of the
operator by relying on Lyapunov analysis. Under a metric subregularity
condition, we further derive a flexible global exponential-type rate for the
distance of the trajectory to the set of fixed points. The results obtained are
applied to analyze the asymptotic behavior of the trajectory of an adaptive
Douglas-Rachford dynamical system, which is applied for finding a zero of the
sum of two operators, one of which is strongly monotone while the other one is
weakly monotone.
