We discuss the long-time dynamical behavior of the non-autonomous suspension bridge-type equation, where the nonlinearity is translation compact and the time-dependent 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:non-autonomous suspension bridge equation; uniform Condition (C); uniform attractor
Consider the following equations:
Suspension bridge equations (1.1) have been posed as a new problem in the field of nonlinear analysis  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 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 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 right-hand 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.
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 [8-10] and the references therein). For the long-time behavior of suspension bridge-type equations, for the autonomous case, in [11,12] 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 .
It is well known that, for a model to describe the real world which is affected by many kinds of factors, the corresponding non-autonomous 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 non-autonomous suspension bridge-type 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 initial-boundary value problem:
where is an unknown function, which could represent the deflection of the road bed in the vertical plane; and are time-dependent 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 non-autonomous dynamics of equation (1.2). At the same time, in mathematics, we only assume that the force term satisfies the so-called Condition () (introduced in ), which is weaker than the assumption of being translation compact (see  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 non-autonomous dynamical system. In Section 3 we prove our main result about the existence of a uniform attractor for the non-autonomous dynamical system generated by the solution of (1.2).
2 Notation and preliminaries
Obviously, we have
Suppose that γ is an arbitrary positive constant, and
where δ is a sufficiently small constant.
By virtue of (2.3), we can get
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 ).
Definition 2.1 ()
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 ()
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 .
Definition 2.3 ()
Theorem 2.4 ()
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
Let X be a Banach space. Consider the space of functions , with values in X that are 2-power integrable in the Bochner sense. is a set of all translation compact functions in , is the set of all translation bound functions in .
In , 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 non-autonomous suspension bridge equation in .
Definition 2.5 ()
Denote by the set of all functions satisfying Condition (). From , 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 ()
In order to define the family of processes of the equations (2.8), we also need the following results:
Proposition 2.8 ()
IfXis reflexive separable, then
Proposition 2.9 ()
3.1 Existence and uniqueness of solutions
At first, we give the concept of solutions for the initial-boundary value problem (2.8).
Then, by using of the methods in  (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,
endowed with the following norm:
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:
3.2 Bounded uniformly absorbing set
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
We assume that ϱ is positive and satisfies
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
We introduce the functional as follows:
Analogous to the proof of Lemma 2.1.3 in , we can estimate the integral and obtain
By virtue of (2.3), we get
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
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
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
Clearly, we get
Combining (3.23)-(3.26), we obtain from (3.22)
We define the functional
By Gronwall’s lemma, we obtain
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 time-dependent external forces only satisfies Condition () instead of translation compact, the uniform attractors in exist.
The authors declare that they have no competing interests.
The authors read and approved the final manuscript.
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 (NWNU-LKQN-11-5).
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