Pullback Attractors for Multivalued Processes and Application to Nonautonomous Problems with Dynamic Boundary Conditions

Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay

“…Then, this guarantees that β p U n is bounded in L p (τ, T ; (V p ) * ), see [18] for more details. Hypotheses (F2), (F3), (9) and recalling that…”

“…After that, some works emerged of this problem, with a nonautonomous term just in perturbations g i can be found in [12,20] and [21], where the authors have established the existence of a uniform attractor and pullback 510 RODRIGO SAMPROGNA AND TOMÁS CARABALLO attractor for the problems, respectively. In [18] the authors considered a nonautonomous term in perturbations f i and ensured the existence of solution as well as the existence of D-pullback attractor for the generalized process associated with a similar problem to (P ) without uniqueness of solution.…”

“…Proof. We will define some appropriate operators to reformulate expression (2) in order to have a simpler functional formulation of our problem, see [7] and [18] for examples of the same method. Then let, for U, V ∈ V p , the following operator…”

“…in the usual way, see [18] for more details. In this way, finding a weak solution of Problem (P ) is equivalent to find a function U with regularities of weak solution definition, and satisfying the following functional equation [18]. In order to find a weak solution to Problem (P ), we use the Faedo-Galerkin approximation.…”

“…see [18] for details. Therefore, U is solution of the limit equation of (11) in the weak star topology of L p (τ, T ; (V p ) * ).…”