Generalized $\phi$-pullback attractors for evolution processes and application to a nonautonomous wave equation

In this work we define the generalized $\phi$-pullback attractors for evolution processes in complete metric spaces, which are compact and positively invariant families, such that they pullback attract bounded sets with a rate determined by a decreasing function $\phi$ that vanishes at infinity. We find conditions under which a given evolution process has a generalized $\phi$-pullback attractor, both in the discrete and in the continuous cases. We present a result for the special case of generalized polynomial pullback attractors, and apply it to obtain such an object for a nonautonomous wave equation.