Strong Uniform Attractors for Nonautonomous Suspension Bridge-Type Equations

We discuss long-term dynamical behavior of the solutions for the nonautonomous suspension bridge-type equation in the strong Hilbert space D A × H2 Ω ∩ H1 0 Ω , where the nonlinearity g u, t is translation compact and the time-dependent external forces h x, t only satisfy condition C∗ instead of translation compact. The existence of strong solutions and strong uniform attractors is investigated using a new process scheme. Since the solutions of the nonautonomous suspension bridge-type equation have no higher regularity and the process associated with the solutions is not continuous in the strong Hilbert space, the results are new and appear to be optimal.


Introduction
Consider the following equations:

1.1
Suspension bridge equations 1.1 have been posed as a new problem in the field of nonlinear analysis 1 by Lazer and McKenna in 1990.This model has been derived as follows.In the suspension bridge system, suspension bridge can be considered as an elastic and unloaded beam with hinged ends.u x, t denotes the deflection in the downward direction; δu t represents the viscous damping.The restoring force can be modeled owing to the cable with one-sided Hooke's law so that it strongly resists expansion but does not resist compression.
The simplest function to model the restoring force of the stays in the suspension bridge can Mathematical Problems in Engineering be denoted by a constant k times u, the expansion, if u is positive, but zero, if u is negative, corresponding to compression; that is, ku , where Besides, the right-hand side of 1.1 also contains two terms: the large positive term l corresponding to gravity, and a small oscillatory forcing term h x, t possibly aerodynamic in origin, where is small.There are many results for 1.1 cf.1-9 , for instance, the existence, multiplicity, and properties of the traveling wave solutions, and so forth.
In the study of equations of mathematical physics, attractor is a proper mathematical concept about the depiction of the behavior of the solutions of these equations when time is large or tends to infinity, which describes all the possible limits of solutions.In the past two decades, many authors have proved the existence of attractor and discussed its properties for various mathematical physics models e.g., see 10-12 and the reference therein .About the long-time behavior of suspension bridge-type equations, for the autonomous case, in 13, 14 the authors have discussed long-time behavior of the solutions of the problem on R 2 and obtained the existence of global attractors in the space H 2 0 Ω × L 2 Ω and D A × H 2 0 Ω .It is well known that, for a model to describe the real world which is affected by many kinds of factors, the corresponding nonautonomous model is more natural and precise than the autonomous one, moreover, it always presents as a nonlinear equation, not just a linear one.Therefore, in this paper, we will discuss the following nonautonomous suspension bridge-type equation: let Ω be an open bounded subset of R 2 with smooth boundary, R τ τ, ∞ , and we add the nonlinear forcing term g u, t which is dependent on deflection u and time t to 1.1 and neglect gravity, then we can obtain the following initial-boundary value problem: where u x, t is an unknown function, which could represent the deflection of the road bed in the vertical plane; h x, t and g u, t are time dependant external forces; ku represents the restoring force, k denotes the spring constant; αu t represents the viscous damping, α is a given positive constant.
To our knowledge, this is the first time to consider the nonautonomous dynamics of 1.3 with the time dependant external forces h x, t and g u, t in the strong topological space D A × H 2 Ω ∩ H 1 0 Ω .At the same time, in mathematics, we only assume that the force term h x, t satisfies the so-called condition C * introduced in 15 , which is weaker than translation compact assumption see 10 or Section 2 below .
This paper is organized as follows.At first, in Section 2, we give recall some preliminaries, including the notation we will use, the assumption on nonlinearity g •, t , and some general abstract results about nonautonomous dynamical system.Then, in Section 3 we prove our main result about the existence of strong attractor for the nonautonomous dynamical system generated by the solution of 1.3 .

Notation and Preliminaries
With the usual notation, we introduce the spaces We equip these spaces with inner product and norm

2.1
Obviously, we have where H * , V * is dual space of H, V , respectively, the injections are continuous, and each space is dense in the following one.
In the following, the assumption on the nonlinearity g is given.Let g be a where G u, s u 0 g w, s dw, and there exists Suppose that γ is an arbitrary positive constant, and where δ is a sufficiently small constant.
As a consequence of 2.3 -2.4 , if we denote G u, s Ω G u, s dx, then there exist two positive constants where m, C 0 > 0, and m is sufficiently small.By virtue of 2.5 , we can get

2.9
When A Δ 2 , problem 1.3 is equivalent to the following equations in H:

2.10
From the Poincaré inequality, there exists a proper constant λ 1 > 0, such that

2.11
We introduce the Hilbert space 12 and endow this space with norm:

2.13
To prove the existence of uniform attractors corresponding to 2.10 , we also need the following abstract results e.g., see 10 . Let where Σ is called the symbol space and σ ∈ Σ is the symbol.
Note that the following translation identity is valid for a general family of processes {U σ t, τ }, σ ∈ Σ, if a problem is the unique solvability and for the translation semigroup {T l | l 0} satisfying T l Σ Σ: A set B 0 ⊂ E is said to be a uniformly w.r.t σ ∈ Σ absorbing set for the family of processes where dim E m m and P m : E → E m is abounded projector.
Theorem 2.4 see 16 .Let Σ be a complete metric space, and let {T t } be a continuous invariant T t Σ Σ semigroup on Σ satisfying the translation identity.A family of processes  where P m : X → X 1 is the canonical projector and δ is a positive constant.
In order to define the family of processes of 2.10 , we also need the following results.

Uniform Attractors in E 1
To describe the asymptotic behavior of the solutions of our system, we set where • 1 denotes the norm in V .

Existence and Uniqueness of Strong Solutions
At first, we give the concept of strong solutions for the initial-boundary value problem 2.10 .
6 and g 0, 0 0. The function z u, u t ∈ L ∞ I; E 1 is said to be a strong solution to problem 2.10 in the time interval I, with initial data z τ for all v ∈ V and a.e.t ∈ I.
Then, by using the methods in 18 Galerkin approximation method , we can get the following result about the existence and uniqueness of strong solutions.Theorem 3.2 existence and uniqueness of strong solutions .Define I τ, T , for all T > τ.
. Then for any given z τ ∈ E 1 , there is a unique solution z u, u t for problem 2.10 in E 1 .Furthermore, for i 1, 2, let {z i τ , h i } z i τ ∈ E 1 and h i ∈ L 2 b R τ ; V be two initial conditions, and denote by z i corresponding solutions to problem 2.10 .Then the estimates hold as follows: for all τ t T τ, Thus, 2.10 will be written as an evolutionary system introduced z t u t , u t t and z τ z τ u 1 , u 2 for brevity, as z 2 , the system 2.10 can be written in the operator form where σ s g u, s , h x, s is the symbol of 3.4 .If z τ ∈ E 1 , then problem 3.4 has a unique solution z t ∈ L ∞ R τ , E 1 .This implies that the process {U σ t, τ } given by the formula U σ t, τ z τ z t is defined in E 1 .

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Now we define the symbol space.A fixed symbol σ 0 s g 0 u, s , h 0 x, s can be given, where h 0 x, s is in L 2 c * R τ ; V , the function g 0 u, s ∈ L 2 c R τ ; M satisfying 2.3 -2.6 , and M is a Banach space, endowed with the following norm: Obviously, the function , where denotes the closure of a set in topological space L 2,w loc R τ ; M or L 2,w loc R τ ; V .So, if g, h ∈ H σ 0 , then g u, t and h x, t all satisfy condition C * .
Applying Propositions 2.8 and 2.9 and Theorem 3.2, we can easily know that the family of processes {U σ t, τ } : E 1 → E 1 , σ ∈ H σ 0 , t τ are defined.Furthermore, the translation semigroup {T l | l ∈ R } satisfies that for all l ∈ R , T l H σ 0 H σ 0 , and the following translation identity: Then for any σ ∈ H σ 0 , the problem 3.4 with σ instead of σ 0 possesses a corresponding to process {U σ t, τ } acting on E 1 .
Consequently, for each σ ∈ H σ 0 , σ 0 s g 0 u, s , h 0 x, s , we can define a process and {U σ t, τ }, σ ∈ H σ 0 is a family of processes on E 1 .

3.11
We can easily see that Then, substituting 3.12 -3.13 into 3.11 , we can obtain that

3.14
In view of 2.6 and 2.8 , we can know

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Consequently,

3.16
We introduce the functional as follows: 3.17 Setting where Analogous to the proof of Lemma 2.1.3 in 10 , we can estimate the integral and obtain

3.20
where m h sup t τ t 1 t m h s ds.By virtue of 2.7 , we can get

3.22
In consideration of 2.9 and 0 < γ < ∞, we can see

3.25
Assume that z τ 2 E 0 R, as t t 0 t 0 B E 0 , we have z t E 0 μ 0 .

3.26
We complete the proof.

A Priori Estimates in E 1
Lemma 3.4.Assuming that z t is a strong solution of 2.10 with initial data then there is a positive constant μ 2 such that for any bounded (in E 1 ) subset B, there exists

3.27
Proof.Now we will prove that z u, u t are bounded in E 1 D A × V .We assume that is positive and satisfies 0 < α − < λ 1 .

3.28
Multiplying 2.10 by Av t Au t t Au t and integrating over Ω, we have 1 2 where A Δ 2 .We can deduce that

3.31
In view of 2.5 and Theorem 3.

3.37
We introduce the functional as follows: Au ku g u, t 3.43

The Existence of Uniform Attractor
We will show the existence of uniform attractor to problem 2.10 in E 1 .Proof.From Theorems 2.4 and 3.5, we merely need to prove that the family of processes

Remark 2 . 6 .
In fact, the function satisfying condition C * implies the dissipative property in some sense, and the condition C * is very natural in view of the compact condition, uniform condition C .Lemma 2.7 see 15 .If f ∈ L 2 c * R; X , then for any > 0 and τ ∈ R we have sup t τ t τ e −δ t−s I − P m f s Definition 2.1 see 10 .Let Σ be a parameter set.{U σ t, τ | t τ, τ ∈ R}, σ ∈ Σ is said to be a family of processes in Banach space E, if for each σ ∈ Σ, {U σ t, τ } is a process; that is, the two-parameter family of mappings {U σ t, τ } from E to E satisfy Let X be a Banach space.Consider the space L 2 loc R; X of functions φ s , s ∈ R with values in X that are 2-power integrable in the Bochner sense.L 2 c R; X is a set of all translation compact functions in L 2 loc R; X , L 2 b R; X is a set of all translation bound functions in L 2 loc R; X .In 15 , the authors have introduced a new class of functions which are translation bounded but not translation compact.In the third section, let the forcing term h x, t satisfy condition C * , we can prove the existence of compact uniform w.r.t.σ ∈ H σ 0 , σ 0 s g 0 u, s , h x, s attractor for nonautonomous suspension bridge equation in E 1 .Definition 2.5 see 15 .Let X be a Banach space.A function f ∈ L 2 b R; X is said to satisfy condition C * if for any > 0, there exists a finite dimensional subspace X 1 of X such that .18 if it i has a bounded uniformly (w.r.t.σ ∈ Σ) absorbing set B 0 ; ii satisfies uniform (w.r.t.σ ∈ Σ) condition (C), where ω τ,Σ B 0 ∩ t τ ∪ σ∈Σ ∪ s t U σ s, t B 0 .Moreover, if E is a uniformly convex Banach space, then the converse is true.m : X → X 1 is the canonical projector.Denote by L 2 c * R; X the set of all functions satisfying condition C * .From 15 , we can see that L Estimates in E 0 Assume that z t is a solution of 2.10 with initial data z0 ∈ B. If the nonlinearity g u, t satisfies 2.3 -2.6 , h 0 ∈ L 2 c * R τ ; H , h ∈ H h 0 , k > 0,then there is a positive constant μ 0 such that for any bounded (in E 0 ) subset B, there exists t 0 t 0 B E 0 such that Proof.Now we will prove that z u, u t are bounded in E 0 V × H.
we choose proper positive constants m and δ, such that 3, we can see that Assuming that V τ E 1 R, as t t 1 t 1 B E 1 , we have And then, combining Theorem 3.2 with Lemma 3.4, we can get the result as follows.Theorem 3.5 bounded uniformly absorbing set inE 1 .Presuming that g 0 ∈ L 2 c R τ ; M and h 0 ∈ L 2 c * R τ ; V .Let g ∈ H g 0 satisfy 2.3 -2.6 , h ∈ H h 0 ,and {U σ t, τ }, σ ∈ H σ 0 H g 0 × H h 0 be the family of processes corresponding to 2.10 in E 1 , then {U σ t, τ } has a uniformly (w.r.t.σ ∈ H σ 0 , and σ 0 g 0 , h 0 , then {U σ t, τ } possesses a compact uniform (w.r.t.σ ∈ H σ 0 ) attractor A H σ 0 in E 1 , which attracts any bounded set in E 1 with • E 1 , satisfying Theorem 3.6 uniform attractor .Let {U σ t, τ } be the family of processes corresponding to problem 2.10 .If g 0 ∈ L 2 c R τ ; M satisfyies 2.3 -2.6 , h 0 ∈ L 2 c * R τ ; V where B 1 is the uniformly (w.r.t.σ ∈ H σ 0 ) absorbing set in E 1 .
, ω 2 , . . ., ω m }, P m : V → V m is an orthogonal projector.For any u, u t ∈ E 1 , we write Since g ∈ L 2 c R τ , M ⊂ L 2 c * R τ , M , h ∈ L 2 c * R τ , H, from Lemma 2.7, we can know for any 1 > 0, there exists a constant m large enough such that Therefore, the family of processes U σ t, τ , σ ∈ H σ 0 satisfies uniformly w.r.t.σ ∈ H σ 0 condition C in E 1 .Applying Theorem 2.4, we can obtain the existence of uniform w.r.t.σ ∈ H σ 0 attractor of the family of processes U σ t, τ , σ ∈ H σ 0 in E 1 , which satisfies 3.44 .We complete the proof.
−1 I − P m h t 2 1 , for t t 1 .e −ω t−s I − P m g u, s 2 M ds, for t t 1 .e −ω t−s I − P m g u, s 2 M ds.