We consider a damped abstract second order evolution equation with an additional
vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we
do not assume the operator defining the main damping to be bounded. First, using a
constructive frequency domain method coupled with a decomposition of frequencies and the
introduction of a new variable, we show that if the limit system is exponentially stable,
then this evolutionary system is uniformly − with respect to the calibration parameter −
exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decay
estimates of the underlying semigroup provided such decay estimates hold for the limit
system. Finally, we discuss some applications of our results; in particular, the case of
boundary damping mechanisms is accounted for, which was not possible in the earlier work
mentioned above.