  The authors consider non-autonomous dynamical behavior of wave-type
evolutionary equations with nonlinear damping and critical nonlinearity. These
type of waves equations are formulated as non-autonomous dynamical systems
(namely, cocycles). A sufficient and necessary condition for the existence of
pullback attractors is established for norm-to-weak continuous non-autonomous
dynamical systems, in terms of pullback asymptotic compactness or pullback
$\kappa-$contraction criteria. A technical method for verifying pullback
asymptotic compactness, via contractive functions, is devised. These results
are then applied to the wave-type evolutionary equations with nonlinear damping
and critical nonlinearity, to obtain the existence of pullback attractors. The
required pullback asymptotic compactness for the existence of pullback
attractors is fulfilled by some new a priori estimates for concrete wave type
equations arising from applications. Moreover, the pullback
$\kappa-$contraction criterion for the existence of pullback attractors is of
independent interest.
