Abstract
We discuss the longtime dynamical behavior of the nonautonomous suspension bridgetype equation, where the nonlinearity is translation compact and the timedependent external forces only satisfy Condition () instead of being translation compact. By applying some new results and the energy estimate technique, the existence of uniform attractors is obtained. The result improves and extends some known results.
MSC: 34Q35, 35B40, 35B41.
Keywords:
nonautonomous suspension bridge equation; uniform Condition (C); uniform attractor1 Introduction
Consider the following equations:
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, the suspension bridge can be considered as an elastic and unloaded beam with hinged ends. denotes the deflection in the downward direction; represents the viscous damping. The restoring force can be modeled owing to the cable with onesided 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 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, , where
Besides, the righthand side of (1.1) also contains two terms: the large positive term l corresponding to gravity, and a small oscillatory forcing term , possibly aerodynamic in origin, where ϵ is small.
There are many results for the problem (1.1) (cf.[17]), for instance, the existence, multiplicity and properties of the traveling wave solutions, etc.
In the study of equations of mathematical physics, the attractor is a proper mathematical concept as regards 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 an attractor and discussed its properties for various mathematical physics models (e.g., see [810] and the references therein). For the longtime behavior of suspension bridgetype equations, for the autonomous case, in [11,12] the authors have discussed longtime behavior of the solutions of the problem on and obtained the existence of global attractors in the space and .
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 a nonlinear equation but not just a linear one. Therefore, in this paper, we will discuss the following nonautonomous suspension bridgetype equation: Let Ω be an open bounded subset of with smooth boundary, , and we add the nonlinear forcing term (which is dependent on the deflection u and time t) to (1.1) and neglect gravity, then we can obtain the following initialboundary value problem:
where is an unknown function, which could represent the deflection of the road bed in the vertical plane; and are timedependent external forces; represents the restoring force, k denotes the spring constant; represents the viscous damping, α is a given positive constant.
To our knowledge, this is the first time for one to consider the nonautonomous dynamics of equation (1.2). At the same time, in mathematics, we only assume that the force term satisfies the socalled Condition () (introduced in [13]), which is weaker than the assumption of being translation compact (see [8] 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 and some general abstract results for a nonautonomous dynamical system. In Section 3 we prove our main result about the existence of a uniform attractor for the nonautonomous dynamical system generated by the solution of (1.2).
2 Notation and preliminaries
With the usual notation, we introduce the spaces , , , where . We equip these spaces with an inner product and a norm , , , and , , respectively,
Obviously, we have
where , is the 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 function from to ℝ and satisfy
where , and there exists , such that
Suppose that γ is an arbitrary positive constant, and
where δ is a sufficiently small constant.
As a consequence of (2.1)(2.2), if we denote , then there exist two positive constants , such that
where , and we can take m sufficiently small.
By virtue of (2.3), we can get
When , the problem (1.2) is equivalent to the following equations in H:
From the Poincaré inequality, there exists a proper constant , such that
We introduce the Hilbert spaces
and endow this space with the norm
To prove the existence of uniform attractors corresponding to (2.8), we also need the following abstract results (e.g., see [8]).
Let E be a Banach space, and let a twoparameter family of mappings on E:
Definition 2.1 ([8])
Let Σ be a parameter set. , is said to be a family of processes in Banach space E, if for each , is a process; that is, the twoparameter family of mappings from E to E satisfy
where Σ is called the symbol space and is the symbol.
Note that the following translation identity is valid for a general family of processes , , if a problem has unique solvability and for the translation semigroup satisfying :
A set is said to be a uniformly (w.r.t. ) absorbing set for the family of processes , if for any and , there exists such that for all . A set is said to be uniformly (w.r.t. ) attracting for the family of processes , , if for any fixed and every ,
Definition 2.2 ([8])
A closed set is said to be the uniform (w.r.t. ) attractor of the family of processes , if it is uniformly (w.r.t. ) attracting (attracting property) and contained in any closed uniformly (w.r.t. ) attracting set of the family of processes , : (minimality property).
Now we recall the results in [14].
Definition 2.3 ([14])
A family of processes , , is said to be satisfying the uniform (w.r.t. ) Condition (C) if for any fixed , and , there exist a and a finite dimensional subspace of E such that
where and is abounded projector.
Theorem 2.4 ([14])
Let Σ be a complete metric space, and letbe a continuous invariantsemigroup on Σ satisfying the translation identity. A family of processes, , possess a compact uniform (w.r.t. ) attractorinEsatisfying
if it
(i) has a bounded uniformly (w.r.t. ) absorbing set; and
(ii) satisfies uniform (w.r.t. ) Condition (C),
where. Moreover, ifEis a uniformly convex Banach space, then the converse is true.
Let X be a Banach space. Consider the space of functions , with values in X that are 2power integrable in the Bochner sense. is a set of all translation compact functions in , is the set of all translation bound functions in .
In [13], the authors have introduced a new class of functions which are translation bounded but not translation compact. In Section 3, let the forcing term satisfy Condition (); we can prove the existence of compact uniform (w.r.t. , ) attractor for a nonautonomous suspension bridge equation in .
Definition 2.5 ([13])
Let X be a Banach space. A function is said to satisfy Condition () if, for any , there exists a finite dimensional subspace of X such that
where is the canonical projector.
Denote by the set of all functions satisfying Condition (). From [13], we can see that .
Remark 2.6 In fact, the function satisfying Condition () implies the dissipative property in some sense, and Condition () is very natural in view of the compact condition, and the uniform Condition (C).
Lemma 2.7 ([13])
whereis the canonical projector andδis a positive constant.
In order to define the family of processes of the equations (2.8), we also need the following results:
Proposition 2.8 ([8])
IfXis reflexive separable, then
(ii) the translation groupis weakly continuous on;
Proposition 2.9 ([8])
(i) for all, , and the setis bound in;
(ii) the translation groupis continuous onwith the topology of;
3 Uniform attractors in
To describe the asymptotic behavior of the solutions of our system, we set and , where denotes the closure of a set in topological space . If , then ; this is to be
3.1 Existence and uniqueness of solutions
At first, we give the concept of solutions for the initialboundary value problem (2.8).
Definition 3.1 Set , for . We suppose that , , satisfying (2.1)(2.4) and . The function is said to be a weak solution to the problem (2.8) in the time interval I, with initial data , provided
Then, by using of the methods in [15] (Galerkin approximation method), we get the following result as regards the existence and uniqueness of solutions:
Theorem 3.2 (Existence and uniqueness of solutions)
Define, . Let, , satisfying (2.1)(2.4). Then for any given, there is a unique solutionfor the problem (2.8) in. Furthermore, for, let (and) be two initial conditions, and denote bythe corresponding solutions to the problem (2.8). Then the estimates hold as follows: for all,
Thus, (2.8) will be written as an evolutionary system, introduced as and for brevity, as , the system (2.8) can be written in the operator form
where is the symbol of (3.3). If , then the problem (3.3) has a unique solution . This implies that the process given by the formula is defined in .
Now we define the symbol space. A fixed symbol can be given, where is in , the function satisfying (2.1)(2.4), and ℳ is a Banach space,
endowed with the following norm:
Obviously, the function is in . We define , where denotes the closure of a set in topological space (or ). So, if , then and all satisfy Condition ().
Applying Proposition 2.8, Proposition 2.9, and Theorem 3.2, we can easily know that the family of processes , , , are defined. Furthermore, the translation semigroup satisfies , , and the following translation identity:
holds.
Then for any , the problem (3.3) with σ instead of possesses a corresponding process acting on .
Consequently, for each , (here , satisfying (2.1)(2.4)), we can define a process
and , , is a family of processes on .
3.2 Bounded uniformly absorbing set
Before we show the existence of bounded uniformly absorbing set, we firstly make a prior estimate of solutions for equations (2.8) in .
Lemma 3.3Assume thatis a solution of (2.8) with initial data. If the nonlinearitysatisfies (2.1)(2.4), , , , then there is a positive constantsuch that for any bounded (in) subsetB, there existssuch that
Proof Now we will prove to be bounded in .
We assume that ϱ is positive and satisfies
Multiplying (2.8) by and integrating over Ω, we have
We easily see that
Then, substituting (3.7)(3.8) into (3.6), we obtain
In view of (2.4) and (2.6), we know
and
Consequently,
We introduce the functional as follows:
Setting , we choose proper positive constants m and δ, such that
Analogous to the proof of Lemma 2.1.3 in [8], we can estimate the integral and obtain
By virtue of (2.3), we get
Choosing , we obtain from (3.13)
In consideration of (2.7) and , we see
and
Combining (3.16), (3.17), and (3.19), we deduce that
We thus complete the proof. □
And then, combining Theorem 3.2 with Lemma 3.3, we get the result as follows.
Theorem 3.4 (Bounded uniformly absorbing set)
Presume thatand. Letsatisfy (2.1)(2.4), , and, be the family of processes corresponding to (2.8) in, thenhas a uniformly (w.r.t. ) absorbing setin. That is, for any bounded subset, there existssuch that
3.3 The existence of uniform attractor
We will show the existence of uniform attractor to the problem (2.8) in .
Theorem 3.5 (Uniform attractor)
Letbe the family of processes corresponding to the problem (2.8). Ifsatisfying (2.1), (2.2), (2.5), and (2.6), , and, thenpossesses a compact uniform (w.r.t. ) attractorin, which attracts any bounded set inwith, satisfying
whereis the uniformly (w.r.t. ) absorbing set in.
Proof From Theorem 2.4 and Theorem 3.4, we merely need to prove that the family of processes , satisfy the uniform (w.r.t. ) Condition (C) in . We assume that , are eigenvalue of operator A in , satisfying
denotes eigenvector corresponding to eigenvalue , , which forms an orthogonal basis in , at the same time they are also a group of canonical basis in V or H, and they satisfy
Let , is an orthogonal projector. For any , we write
Choosing and . Taking the scalar product with for (2.8) in H, we have
where
Clearly, we get
Combining (3.23)(3.26), we obtain from (3.22)
We define the functional
By Gronwall’s lemma, we obtain
Obviously, there exists a constant , such that
so
Since , , from Lemma 2.7, we can know for any , there exists a constant m large enough such that
Therefore, the family of processes , satisfy uniformly (w.r.t. ) Condition (C) in . Applying Theorem 2.4, we can obtain the existence of a uniform (w.r.t. ) attractor of the family of processes , in , which satisfies (3.21).
We thus complete the proof. □
So we can draw the conclusion: when the nonlinearity is translation compact and the timedependent external forces only satisfies Condition () instead of translation compact, the uniform attractors in exist.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors read and approved the final manuscript.
Acknowledgements
This work is partly supported by NSFC (11361053, 11201204, 11101134, 11261053, 11101404) of China and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University (NWNULKQN115).
References

Lazer, AC, McKenna, PJ: Largeamplitude periodic oscillations in suspension bridges: some new connection with nonlinear analysis. SIAM Rev.. 32(4), 537–578 (1990). Publisher Full Text

An, Y, Zhong, C: Periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load. J. Math. Anal. Appl.. 279, 569–579 (2003). Publisher Full Text

Choi, QH, Jung, T: A nonlinear suspension bridge equation with nonconstant load. Nonlinear Anal.. 35, 649–668 (1999). Publisher Full Text

Humphreys, LD: Numerical mountain pass solutions of a suspension bridge equation. Nonlinear Anal. TMA. 28(11), 1811–1826 (1997). Publisher Full Text

Lazer, AC, McKenna, PJ: Large scale oscillatory behavior in asymmetric systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 4, 243–274 (1987)

Ma, Q, Zhong, C: Existence of global attractors for the coupled system of suspension bridge equations. J. Math. Anal. Appl.. 308, 365–379 (2005). Publisher Full Text

McKenna, PJ, Walter, W: Nonlinear oscillation in a suspension bridge. Nonlinear Anal.. 39, 731–743 (2000). Publisher Full Text

Chepyzhov, VV, Vishik, MI: Attractors for Equations of Mathematics and Physics, Am. Math. Soc., Providence (2002)

Hale, JK: Asymptotic Behavior of Dissipative Systems, Am. Math. Soc., Providence (1988)

Temam, R: InfiniteDimensional Dynamical Systems in Mechanics and Physics, Springer, New York (1997)

Ma, Q, Zhong, C: Existence of global attractors for the suspension bridge equations. J. Sichuan Univ.. 43(2), 271–276 (2006)

Zhong, C, Ma, Q, Sun, C: Existence of strong solutions and global attractors for the suspension bridge equations. Nonlinear Anal.. 67, 442–454 (2007). Publisher Full Text

Ma, S, Zhong, C: The attractors for weakly damped nonautonomous hyperbolic equations with a new class of external forces. Discrete Contin. Dyn. Syst.. 18, 53–70 (2007)

Lu, S, Wu, H, Zhong, C: Attractors for nonautonomous 2D NavierStokes equations with normal external forces. Discrete Contin. Dyn. Syst.. 13, 701–719 (2005)

Borini, S, Pata, V: Uniform attractors for a strongly damped wave equations with linear memory. Asymptot. Anal.. 20, 263–277 (1999)