Abstract
We discuss the longtime dynamical behavior of the nonautonomous suspension bridgetype
equation, where the nonlinearity
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.
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
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
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
where
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
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
2 Notation and preliminaries
With the usual notation, we introduce the spaces
Obviously, we have
where
In the following, the assumption on the nonlinearity g is given. Let g be a
where
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
where
By virtue of (2.3), we can get
When
From the Poincaré inequality, there exists a proper constant
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
Definition 2.1 ([8])
Let Σ be a parameter set.
where Σ is called the symbol space and
Note that the following translation identity is valid for a general family of processes
A set
Definition 2.2 ([8])
A closed set
Now we recall the results in [14].
Definition 2.3 ([14])
A family of processes
(i)
(ii)
where
Theorem 2.4 ([14])
Let Σ be a complete metric space, and let
if it
(i) has a bounded uniformly (w.r.t.
(ii) satisfies uniform (w.r.t.
where
Let X be a Banach space. Consider the space
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
Definition 2.5 ([13])
Let X be a Banach space. A function
where
Denote by
Remark 2.6 In fact, the function satisfying Condition (
Lemma 2.7 ([13])
If
where
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
(i) for all
(ii) the translation group
(iii)
Proposition 2.9 ([8])
Let
(i) for all
(ii) the translation group
(iii)
3 Uniform attractors in
E
0
To describe the asymptotic behavior of the solutions of our system, we set
where
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 all
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
Thus, (2.8) will be written as an evolutionary system, introduced as
where
Now we define the symbol space. A fixed symbol
endowed with the following norm:
Obviously, the function
Applying Proposition 2.8, Proposition 2.9, and Theorem 3.2, we can easily know that
the family of processes
holds.
Then for any
Consequently, for each
and
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 that
Proof Now we will prove
We assume that ϱ is positive and satisfies
Multiplying (2.8) by
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
hold, then
We define
where
Analogous to the proof of Lemma 2.1.3 in [8], we can estimate the integral and obtain
where
By virtue of (2.3), we get
Choosing
In consideration of (2.7) and
and
Combining (3.16), (3.17), and (3.19), we deduce that
Assuming 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 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)
Let
where
Proof From Theorem 2.4 and Theorem 3.4, we merely need to prove that the family of processes
Let
where
Choosing
where
Clearly, we get
Combining (3.23)(3.26), we obtain from (3.22)
We define the functional
and we set
By Gronwall’s lemma, we obtain
Obviously, there exists a constant
so
Since
where
Let
So for every
where
Therefore, the family of processes
We thus complete the proof. □
So we can draw the conclusion: when the nonlinearity
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).
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