International Journal of Modern Nonlinear Theory and Application
Vol.06 No.04(2017), Article ID:81233,14 pages
10.4236/ijmnta.2017.64012
Random Attractors for the Kirchhoff-Type Suspension Bridge Equations with Strong Damping and White Noises
Chuangliang Qin1, Jinji Du1, Guoguang Lin2*
1School of Mathematics and Information, Xinyang College, Xinyang, China
2School of Mathematics and Statistics, Yunnan University, Kunming, China
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: October 28, 2017; Accepted: December 18, 2017; Published: December 21, 2017
ABSTRACT
In this paper, we investigate the existence of random attractor for the random dynamical system generated by the Kirchhoff-type suspension bridge equations with strong damping and white noises. We first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the existence of the global attractors of the equation.
Keywords:
Kirchhoff-Type Suspension Bridge Equations, Random Attractors, Random Dynamical System
1. Introduction
In this paper, we consider the following stochastic Kirchhoff-type suspension bridge equations
(1.1)
where is an unknown function,which represents the downward deflection of the road bed in the vertical plane, for and for . denotes the spring constant of the ties, the real constant p represents the axial force acting at the end of the road bed of the bridge in the reference configuration. Namely, p is negative when the bridge is stretched, positive when compressed. is an open bounded subset of with sufficiently smooth boundary . is not identically equal to zero, is a nonlinear function satisfying certain conditions. is the derivative of a one-dimensional two-valued Wiener process and formally describes white noise.
We assume that the nonlinear function with , which satisfies the following assumptions:
(a) Growth conditions:
(1.2)
where is a positive constant. For example, obviously, satisfies (1.2).
(b) Dissipation conditions:
(1.3)
and
(1.4)
where are positive constants.
When and , Equation (1.1) is regarded as a model of naval structures,which is originally in [1] introduced by Lazer and McKenna. To the best of our knowledge, Qin [2] [3] proved random attractor for stochastic Kirchhoff equation with white noise, Ma [4] investigated the asymptotic behavior of the solution for the floating beam, that is, the “noise” is absent in (1.1). No one else has studied the long-time behavior of the solutions about these problems, it is just our interest in this paper. As far as the other related problems are concerned, we refer the reader to [2] - [7] and the references therein.
It is well known that Crauel and Flandoli originally introduced the random attractor for the infinite-dimensional RDS [8] [9] . A random attractor of RDS is a measurable and compact invariant random set attracting all orbits. It is the appropriate generalization of the now classical attractor exists, it is the smallest attracting compact set and the largest invariant set [10] . Zhou et al. [11] studied random attractor for damped nonlinear wave equation with white noise. Fan [12] proved random attractor for a damped stochastic wave equation with multiplicative noise. These abstract results have been successfully applied to many stochastic dissipative partial differential equations. The existence of a random attractors for the wave equations has been investigated by several authors [8] [9] [10] .
The outline of this paper is as follows: In Section 2, we recall many basic concepts related to a random attractor for genneral random dynamical system. In Section 3, We prove the existence and uniqueness of the solution corresponding to system (1.1) which determines RDS. In Section 4, we prove the existence of random attractor of the random dynamical system.
2. Random Dynamical System
In this section, we recall some basic concepts related to RDS and a random attractor for RDS in [8] [9] [10] , which are important for getting our main results.
Let be a separable Hilbert space with Borel s-algebra , and let be a probability space. is a family of measure preserving transformations such that is measurable, and for all . The flow together with the probability space is called a metric dynamical system.
Definition 2.1. Let be a metric dynamical system. Suppose that the mapping is -measurable and satisfies the following properties:
1) and ;
2) , for all and .
Then is called a random dynamical system (RDS). Moreover, is called a continuous RDS if is continuous with respect to x for and .
Definition 2.2. A set-valued map is said to be a closed (compact)random set if is closed (compact) for , and is measurable for all .
Definition 2.3. If K and B are random sets such that for there exists a time such that for all ,
then K is said to absorb B, and is called the absorption time.
Definition 2.4. A random set is called a random attractor associated to the RDS if :
1) is a random compact set, i.e., is compact for , and the map is measurable for every ;
2) is -invariant, i.e., for all and ;
3) attracts every set B in X, i.e., for all bounded (and non-random) ,
where denotes the Hausdorff semi-distance:
Note that can be interpreted as the position of the trajectory which was in x at time . Thus, the attraction property holds from .
Theorem 2.1. [8] (Existence of a random attractor) Let be a continuous random dynamical system on X over . Suppose that there exists a random compact set absorbing every bounded non-random set . Then the set
is a global random attractor for , where the union is taken over all bounded , and is the w-limitsset of B given by
3. Existence and Uniqueness of Solutions
With the usual notation, we denote
where . We denote with the following inner products and norms,respectively:
And we introduce the space , which is used throughout the paper and endow the space E with the following usual scalar product and norm:
where T denotes the transposition.
More generally, define for , which turns out to be a Hilbert space with the inner product , we denote by the norm on induced by the above inner product. Let
be the first eigenvalue of , by the compact embeddings along with the generalized Poincaré inequality, we have
(3.1)
It is convenient to reduce (1.1) to an evolution of the first order in time
(3.2)
whose equivalent Itó equation is
(3.3)
where is a one-dimensional two-sided real-valued Wiener process on . Without loss of generality,we can assume that
where P is a Wiener measure. We can define a family of measure preserving and ergodic transformations by
Let , we consider the random partial differential equation equivalent to (3.3)
(3.4)
Apparently, there is no stochastic differential in (3.4) by comparing with stochastic differential Equation (3.3). Let
then (3.4) can be written as
(3.5)
From [13] we know that L is the infinitesimal generators of -semigroup on E. It is not difficult to check that the functions is locally Lipschitz continuous with respect to and bounded for every . By the classical semigroup theory of existence and uniqueness of solutions of evolution differential equations [13] , so we have the following theorem:
Theorem 3.1. Consider (3.5). For each and initial value , there exists a unique function such that satisfies the integral equation
By theorem 3.1, we can prove that for every the following statements hold for all :
1) If , then .
2) is continuous in t and .
3) The solution mapping of (3.5) satisfies the properties of RDS.
Equation (3.5) has a unique solution for every . Hence the solution mapping
(3.6)
generates a random dynamical system, so the transformation
(3.7)
also determines a random dynamical system corresponding to Equation (3.2).
4. Existence of a Random Attractor
In this section,we prove the existence of a random attractor for RDS (3.7) in E. Let , where
(4.1)
So Equation (3.4) can be written as
(4.2)
where
The mapping
is defined by (4.2).
To show the conjugation of the solution of the stochastic partial differential Equation (1) and the random partial differential Equation (4.2), we introduce the homeomorphism
with the inverse homeomorphism . Then the transformation
(4.3)
also determines RDS corresponding to Equation (1). Therefore, for RDS (7) we only need consider the equivalent random dynamical system , where is decided by
(4.4)
where
Next, we prove a positivity property of the operator Q in E that plays a vital role throughout the paper.
Lemma 4.1. For any , there holds
Proof. Since , by using the Poincaré inequality and the Young inequality, we conclude that
where . ,
Lemma 4.2. Let (1.2)-(1.4) hold, there exist a random variable , and a bounded ball of centered at 0 with random radius such that for any bounded non-random set of , there exists a deterministic such that the solution of (4.2) with initial value satisfies for ,
and for all
(4.5)
where , and is given by
Besides it is easy to deduce a similar absorption result for
instead of .
Proof. We take the inner product in E of (4.2) with , where , we get
(4.6)
where
(4.7)
We deal with the terms in (4.7) one by one as follows:
(4.8)
(4.9)
(4.10)
(4.11)
By using (1.2)-(1.3) and the Hölder inequality, we get
(4.12)
Inequality (4.12) together with (1.4) yields
(4.13)
(4.14)
(4.15)
Collecting with (4.6)-(4.15) and Lemma 4.1, we get that
where , , and
By the Gronwall lemma, we conclude that
(4.16)
Let
where and are finite , we get a bounded set B of E, we choose such that
(4.17)
for all , and
(4.18)
for all , and for all . ,
Let be a solution of problem (1.1) with initial value . we make the decomposition , where and satisfy
(4.19)
and
(4.20)
Lemma 4.3. Let , be a bounded non-random subset of ,
(4.21)
where satisfies (4.19),
Proof. Let , taking the scalar product in H of (4.19), we get that
(4.22)
using the Hölder inequality and the Young inequality, we get that
(4.23)
we have that
Let , we can get that
and for we get
so
By the Gronwall lemma, getting that (4.21). ,
Lemma 4.4. Assume that (1.2) holds, there exists a random radius , such that for ,
(4.24)
where satisfies (4.20).
Proof. Let , Equation (4.20) can be written to
(4.25)
where
Taking the scalar product in of (4.25) with , we get that
(4.26)
where
(4.27)
Due to Lemma 4.1, we get that
(4.28)
Using the Young inequality, we have that
(4.29)
(4.30)
(4.31)
(4.32)
By (1.2), (4.5) and Sobolev embedding theorem, we obtain that is uniformly bounded in , that is, there exists a constant such that
(4.33)
Combining with (4.33), the Sobolev embedding theorem and the Young inequality, we have that
(4.34)
where is a positive constant.
(4.35)
Let , by the Poincaré and , we get
that, . Using (4.27)-(4.35) and (4.5), for , from (4.27) we get that
Using the Gronwall lemma, we get that
(4.36)
Set
Since , is finite P-a.s., together with 4.18 and 4.36, we get that
This completes the proof of Lemma 4.4. ,
Theorem 4.5. Let , (1.2)-(1.4) hold, , then the
random dynamical system possesses a nonempty compact random attractor .
Proof. Let be the ball of of radius
, by the compact embedding , it follows that is
compact in E. for every bounded non-random set B of E and any , by Lemma 4.4, we know that . Therefore, for ,
So, for all ,
From relation (4.3) between and , w can obtain that for any non-random bounded ,
Hence, the RDS associated with (3.7) possesses a uniformly attracting compact set . Using Theorem 2.1, we complete the proof.
Acknowledgements
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11561076.
Cite this paper
Qin, C.L., Du, J.J. and Lin, G.G. (2017) Random Attractors for the Kirchhoff-Type Suspension Bridge Equations with Strong Damping and White Noises. International Journal of Modern Nonlinear Theory and Application, 6, 134-147. https://doi.org/10.4236/ijmnta.2017.64012
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