Strong Pullback Attractors for Nonautonomous Suspension Bridge Equations

μu t represents the viscous damping, and μ is a given positive constant. Suspension bridge equations have been posed as a new problem in the field of nonlinear analysis [1] by Lazer and Mckenna in 1990. There are many results for the problem (1) (cf. [1–8]), for instance, the existence, multiplicity, and properties of the travelling wave solutions, and so forth. About the long-time behavior of suspension bridge equations, for the autonomous case, in [9, 10], the authors have discussed long-time behavior of the solutions of the problem on R2 and obtained the existence of global attractors in the space

Choosing  = min{ 1 ,  2 1 }, by the Poincaré inequality, we have We introduce the Hilbert spaces and endow this space with norm This paper is organized as follows.At first, in Section 2, we recall some preliminaries and results concerning the pullback attractor.Then, in Section 3, we prove our main result about the existence of pullback D-attractor for the nonautonomous dynamical system generated by the solution of (1).
Let P() denote the family of all nonempty subsets of , let B() be the set of all bounded subsets of , and let K be the class of all families D = {  } ∈ ⊂ P().We consider a nonempty subclass D ∈ K.
We need the following lemmas in order to prove the main result.

Pullback D-Attractors for Nonautonomous Suspension Bridge Equations
First, we give the following result.

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Using the Hölder and Young inequalities, we obtain We can easily see that Set Thus, denote (33) By the Gronwall lemma, we have Set From Theorem 8, we have where ISRN Applied Mathematics and then Set Let  = {()} ∈R ∈  ,E 1 .Combining (36) and (39), we have ) for all  0 ∈ ( − ),  ∈ R, and  ⩾ 0. Set and consider the family B,E 1 of closed balls in E 1 defined by From ( 18) and (41), B,E 1 is a pullback D ,E 1 -absorbing for the cocycle  in E 1 .
Next, we show that the cocycle  satisfies the pullback D ,E 1 -Condition ().