Generalized \(\varphi \)-Pullback Attractors for Evolution Processes and Application to a Nonautonomous Wave Equation

In this work we define the generalized \(\varphi \) $\phi$ -pullback attractors for evolution processes in complete metric spaces, which are compact and positively invariant families, that pullback attract bounded sets with a rate determined by a decreasing function \(\varphi \) $\phi$ that vanishes at infinity, called decay function. We find conditions under which a given evolution process has a generalized \(\varphi \) $\phi$ -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.