Strong Pullback Attractors for Nonautonomous Suspension Bridge Equations

We prove the existence of a pullback -attractor in for the nonautonomous suspension bridge equations. 1. Introduction In this paper, we consider the following nonautonomous suspension bridge equation: where is a bounded domain of with a smooth boundary , is an unknown function, which could represent the deflection of the road bed in the vertical plane, represents the restoring force, denotes the spring constant, represents the viscous damping, and 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 and obtained the existence of global attractors in the space and . Caraballo et al. advanced the concept of the pullback -attractor in [11], and the existence of the pullback attractors was proved under the assumptions of asymptotic compactness and existence of a family of absorbing sets. Recently, Park and Kang [12] studied the pullback -attractor for suspension bridge equations in the weak space . Motivated by the ideas of [11, 13], we study the existence of a strong pullback -attractor for the nonautonomous suspension bridge equations in the strong topological space . The nonlinear functions satisfy the following assumptions: where constant . With the usual notation, we introduce the spaces , , where . We equip these spaces with inner product and norm？？ , , and , respectively: Obviously, we have where is dual space of , respectively; the injections are continuous and each space is dense in the following one. Choosing , 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 -attractor for the nonautonomous dynamical system generated by the solution of (1). 2. Notation and Preliminaries Let be a complete metric space, be a metric space which will be called the parameter space. We define a nonautonomous dynamical system by a cocycle mapping which is driven by an autonomous dynamical system acting on a parameter space . Specifically, is a dynamical system on ; that