Research Article | Open Access
Suping Wang, Qiaozhen Ma, "Strong Pullback Attractors of the Nonautonomous Suspension Bridge Equation with Variable Delay", Mathematical Problems in Engineering, vol. 2019, Article ID 1930369, 12 pages, 2019. https://doi.org/10.1155/2019/1930369
Strong Pullback Attractors of the Nonautonomous Suspension Bridge Equation with Variable Delay
In this paper, we consider the suspension bridge equation with variable delay. The long-time dynamics of the solutions for the suspension equations without delay effects have been investigated by many authors. But there are few works on suspension equations with delay. Moreover there are not many studies on attractors for other systems with delay. Thus, we study the existence of pullback attractor for the suspension equation with variable delay by using the theory of attractors for multivalued dynamical systems.
The suspension bridge equations were presented by Lazer and McKenna in  as a new problem in the field of nonlinear analysis. Ma and Zhong et al. have investigated systematically the long-time behavior of solution for both the single and the coupled suspension bridge equations in [2–6]. Bochicchio, Giorgi, and Vuk studied the existence of the global attractor for the Kirchhoff suspension bridge equation and obtained a regularity result of attractor; see  for details. The existence of global attractors for the suspension bridge with linear memory was achieved by Kang in . However, none of the above conclusions contain time delay. In regard to wave equation with delay, Wang in  studied the wave equation with variable delay; they derived the existence of pullback attractors under some assumptions on delay term. For the related works of equations with variable delay, we also refer to [10–13] and the references therein. Motivated by the above mentioned works, our goal is to prove the existence of pullback attractor for suspension bridge with variable delay. To the best our knowledge, there is no results on suspension bridge equation with variable delay. Here we study the existence of pullback attractors for the suspension bridge equation with variable delay. That is, we consider the following nonautonomous suspension bridge equations in :
Model (1) describes the vibration of the road bed in the vertical direction, where denotes the deflection in the downward direction, represents the viscous damping, and is a positive constant. represents the restoring force, denotes the spring constant, the function . is the initial time, is the initial datum on the interval , where , and source term contains some hereditary characteristic and satisfies the following conditions:
(I) The functions and , and satisfies
(II) satisfies where will be determined later.
We will denote by the Banach space , equipped with the sup-norm. For an element , its norm will be written as .
Let be two Banach spaces with continuous compact embedding and denote Banach space . For an element , its norm will be written as Also, we will use the spaces and in our analysis. Given and , for each , we denote by the function defined on by the relation .
The rest of this article consists of three sections. In the next section, we define some functions setting and iterate some useful lemmas and abstracts. In Section 3, we show the weak and stronger weak solutions. In Sections 4 and 5, we prove the existence of pullback attractor for (1) in , , respectively.
In this section, we iterate some notations and abstract results.
We define Hilbert space ; the scalar product and the norm are denoted bywhere . When , denote ; when , , the scalar product and the norm on and are denoted as follows:
Define ; here . In particular, we denote as the norm on . It is obvious that ; here are the dual space of respectively, and each space is dense in the following one and the injections are continuous. By the Poincaré inequality, we have where is the first eigenvalue of .
We first need the following definitions and abstract results about pullback attractors.
Let be a complete metric space with metric ; we denote by and , respectively, the Hausdorff semidistance and Hausdorff distance between two nonempty subsets of a complete metric space , which are defined bywhere andFinally, denote by the open neighborhood of radius of subset of .
Let be a nonempty class of parameterized sets , and be a class of nonempty closed subsets of .
Definition 1 (see ). A family of mappings , is called a multivalued process if it satisfies(1);(2).
A collection of some families of nonempty closed subsets of is said to be inclusion-closed if for each , is a nonempty subset of and also belongs to ; see .
Let be a multivalued process on . is called a pullback absorbing set for if, for any and each , there exists a such that
Definition 2 (see ). A family of nonempty compact subsets is called to be a pullback attractor for the multivalued process if it satisfies the following:(1) is invariant; i.e., (2) attracts every member of ; that is, for every and any fixed ,
Definition 3 (See ). Let be a multivalued process on . is said to be pullback asymptotically upper-semicompact in with respect to if, for any fixed , any sequence has a convergent subsequence in whenever with .
Lemma 4 (see [15, 16]). Let and be two Banach spaces such that with a continuous injection. If a function belongs to and is weakly continuous with values in , then is weakly continuous with values in .
Theorem 5 (see ). Let be a multivalued process on Banach space , and let be a pullback absorbing set for in . Suppose that can be written as and for any fixed ,(1);(2)for any fixed , every sequence is a Cauchy sequence in .
Then is pullback asymptotically upper-semicompact in .
Theorem 6 (see ). Let be an inclusion-closed collection of some families of nonempty closed subsets of and be a multivalued process on . Also has closed values, and let be upper-semicontinuous in for fixed . Suppose that is pullback asymptotically upper-semicompact in , has a pullback absorbing set in , and is closed for every . Then, the pullback attractor is unique and is given by
In this section, we will construct the multivalued processes associated with our problem (1), and in the sequel denotes arbitrary positive constants, which may be different from line to line and even in the same line. For convenience, let us define the following Hilbert space:
Theorem 7. Suppose that holds true and . Then for each , it follows that
(i) for any , there exists a solution corresponding to (1) and satisfies(ii) If in addition and the function , then for any , problem (1) admits a stronger weak solution
Proof. We exploit the Galerkin Approximation method. First, we assume that there exists an orthogonal basis of consisting of eigenvectors of in ; the corresponding eigenvalues are , satisfyingFor each , according to the basic theory of ordinary differential equations, there exists an approximate solution of the formand it satisfieswhere is the orthogonal projector in , and . We next derive a priori estimate for the Galerkin approximate solutions.
Multiplying (20) by yieldsNote that . We choose small enough such that , and find thatand Exploiting Young inequality, we can see thatand in equality (25), we use , where . Taking and then combining with (22)–(25), from (21) we get the following:Setthen Integrating (28) from to , we getExploiting assumption , let . Note that and, for , we arrive atThus, together with (30), we getBy Gronwall’s lemma, we can deduce thatThus we can easily infer from (32) that Thus there exists a subsequence, still denoted by and , such thatandBy passing to the limit in (20) we immediately obtain that is a solution of (1) which satisfies .
Moreover, similar to the proof of Theorem 4.1 in  and Lemma 4, we obtainfurthermore,Next, we prove the existence of stronger weak solution.
Assume that there exists an orthonormal basis of consisting of eigenvectors of in . The corresponding eigenvalues are , satisfying the following.For each , according to the basic theory of ordinary differential equations, there exists an approximate solution of the form and it satisfieswhere is the orthogonal projector in , and .
Take the inner product in of (20) with to getBy using the Hölder and Young’s inequalities, we have the estimatesand, by using Young’s inequality, we haveand Choose small enough, such that . Combining with (42)–(44), from (41),Set the following.Taking , then (45) can be rewritten asIntegrating (47) from to , we obtainExploiting (32), we getso we can deduce thatThus we can infer from (50) that are uniformly bounded in and , respectively, as . Combining with the results of (1) in Theorem 7 and exploiting Theorem 4.1 in  Sec. IV.4.4, by a standard argument we obtain the existence of strong solutions in . We can also refer to .
4. Pullback Attractors in
4.1. Pullback Absorbing Set in
Owing to Theorem 7, we can define a family of multivalued mappings as follows: ;
is a solution of (1) as in with and ;
is a solution of (1) as in with , where is defined for as . Then are multivalued processes on and , respectively.
we will denote by all the families of nonempty subsets such that where will be determined later.
Lemma 8. Suppose that hold true. Then the multivalued process corresponding to (1) possesses a pullback absorbing set in .
Let , and then using Gronwall lemma to (58), we arrive atFurthermore, let , such that , which yields
Now setting instead of (where ), there holdsLet , exploiting and (61) implies that