Uniform attractors for non-autonomous suspension bridge-type equations
© Wang et al.; licensee Springer. 2014
Received: 5 July 2013
Accepted: 18 March 2014
Published: 28 March 2014
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.
Keywordsnon-autonomous suspension bridge equation uniform Condition (C) uniform attractor
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 .
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
where , is the dual space of H, V, respectively, the injections are continuous and each space is dense in the following one.
where δ is a sufficiently small constant.
where , and we can take m sufficiently small.
To prove the existence of uniform attractors corresponding to (2.8), we also need the following abstract results (e.g., see ).
Definition 2.1 ()
where Σ is called the symbol space and is the symbol.
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 ()
is bounded; and
where and is abounded projector.
Theorem 2.4 ()
has a bounded uniformly (w.r.t. ) absorbing set ; and
satisfies uniform (w.r.t. ) Condition (C),
where . Moreover, if E is 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 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 ()
where is the canonical projector.
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 ()
where is the canonical projector and δ is 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 ()
for all , ;
the translation group is weakly continuous on ;
for all .
Proposition 2.9 ()
for all , , and the set is bound in ;
the translation group is continuous on with the topology of ;
for all .
3 Uniform attractors in
where denotes the norm in H.
3.1 Existence and uniqueness of solutions
At first, we give the concept of solutions for the initial-boundary value problem (2.8).
for all and a.e. .
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)
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 .
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 ().
Then for any , the problem (3.3) with σ instead of possesses a corresponding process acting on .
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 .
Proof Now we will prove to be bounded in .
hold, then .
where , .
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)
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)
where is the uniformly (w.r.t. ) absorbing set in .
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.
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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