Asymptotic Behavior of the Kirchhoff Type Stochastic Plate Equation on Unbounded Domains

Abstract

In this paper, we study the asymptotic behavior of solutions to the Kirchhoff type stochastic plate equation driven by additive noise defined on unbounded domains. We first prove the uniform estimates of solutions and then establish the existence and upper semicontinuity of random attractors.

1. Introduction

Plate equations can be found in many fields such as certain physical areas as to vibration and elasticity theories of solid mechanics. In this paper, we consider the following Kirchhoff type stochastic plate equation with additive noise defined onwhere , with , and are positive constants, is a nonlinearity satisfying certain growth and dissipative conditions, and are given functions in and , respectively, and is a two-sided real-valued Wiener process on a probability space.

The function satisfies the following conditions:(1), such that where and are some positive real constants.(2)Let ; for ,

For the nonlinear function , we presume and let ; for and , there exist positive constants , such thatwhere . Note that (4) and (5) imply

As for deterministic plate equations, many authors have showed the existence of global attractors (see [1–9]). For the stochastic case, the existence of random attractors for plate equations has been investigated in [10, 11, 12] on bounded domains. In addition, there are results about the existence of random attractors and asymptotic compactness for plate equations on unbounded domains in [13–18].

When in (1), we have investigated the existence of a random attractor for plate equations with additive noise and nonlinear damping defined on (see [14]). However, when equation (1) is Kirchhoff type, the problem is not yet considered by any predecessors.

To overcome the noncompactness of Sobolev embeddings on , we will apply the idea of uniform estimates on the tails of solutions as in [19, 20] as well as the compactness methods introduced in [21]. More precisely, we first show that the tails of the solutions of (1) are uniformly small outside a bounded domain for large time, and then we derive the asymptotic compactness of solutions in bounded domains by splitting the solutions.

This paper is organized as follows. In Section 2, we present some notations and proposition about random dynamical systems. In Section 3, we define a continuous cocycle for equation (1) in . In Section 4, we obtain all necessary uniform estimates of solutions. In Section 5, we show the existence and uniqueness of a random attractor for (1) defined on .

2. Notations

In this section, we present some basic notations and known results on nonautonomous random dynamical systems which can be found in [22, 20].

Let be a complete separable metric space and be an ergodic metric dynamical system (see [23]).

Proposition 1. Let be an inclusion closed collection of some families of nonempty subsets of and be a continuous cocycle on over . Then, has a unique -pullback random attractor in if is -pullback asymptotically compact in and has a closed measurable -pullback absorbing set in .
Next, we present criteria concerning upper semicontinuity of nonautonomous random attractors with respect to a parameter.

Theorem 1. Let be a separable Banach space and be an autonomous dynamical system with the global attractor in . Given , suppose that is the perturbed random dynamical system with a random attractor and a random absorbing set . Then, for ,if the following conditions are satisfied:(i)There exists some deterministic constant such that, for ,(ii)There exists , such that for ,(iii)For , , and with , it holds that where .

3. Cocycles for Stochastic Plate Equation

In this section, we present some basic settings about (1) and prove that it generates a continuous cocycle in .

Let denote the Laplace operator in , , with the domain . We can define the powers of for . The space is a Hilbert space with the following inner product and norm.

Set with normfor , where stands for the transposition.

Let , where is a small positive constant whose value will be determined later; then, (1) is equivalent towhere .

For : we assume that there exists a positive constant such thatandwhere denotes the absolute value of real number in .

Denote ; then, we consider the Ornstein–Uhlenbeck equation and the Ornstein–Uhlenbeck process

From [24], it is known that the random variable is tempered, and there is a -invariant set of full measure such that is continuous in for every . Putwhich solves

Lemma 1 (see [25]). For any , there exists a tempered random variable , such that for all ,where satisfiesNow, let , and we havewhere . We will consider (23) for and write as from now on.
The well-posedness of the deterministic problem (23) in can be established by standard methods as in [13, 26–28]. If (2)–(7) are fulfilled, let , where . Then, for every and , problem (23) has a unique -measurable solution with being continuous in for each . Moreover, for every , the mappinggenerates a continuous cocycle from to over and .

4. Uniform Estimates of Solutions

In this section, we derive uniform estimates on the solutions of problem (23) and construct a tempered pullback absorbing set.

Let be small enough such thatand define appearing in (17) by

Lemma 2. Assume that (2)–(7) and (16) hold. Then, for every , and , there exists such that for all ,where and is a positive constant depending on , and but independent of , and .

Proof. Taking the inner product of with in , we find thatBy , we getNext, we estimate some terms of (28).By (5), we getBy conditions (4) and (6), we obtainUsing the Cauchy–Schwarz inequality and the Young inequality, there holdsBy (2) and (3) we haveBy (30)–(38), it follows from (28) thatBy (14) and (26), we get from (39) thatMultiplying (40) by and integrating over and then replacing by , we getIt follows from Lemma 1 thatFrom (8),Because and , we get from (43) thatTherefore, (17), (42), and (43) deduce the desired result (27).

Lemma 3. Assume that (2)–(7) and (16) hold. Then, there exists a random ball centered at 0 with random radiussuch that is a closed measurable -pullback absorbing set for the continuous cocycle associated with problem (23) in , that is, for every , and , there exists , such that for all ,

Proof. This is an immediate consequence of (24) and Lemma 2.
Let be a smooth function such that for all , andFor every , letThen, there exist positive constants , and independent of such that , , , for all and .
Given, denote and be the complement of. To prove asymptotic compactness of solution on , we prove the following lemma.

Lemma 4. Assume that (2)–(7) and (16) hold. Then, for every , and , there exist and , such that for all ,where .

Proof. Taking the inner product of with in , we obtainUsing the Young inequality and the Sobolev interpolation inequality,We haveBy (6), we getBy (4) and (6), we getFor the remainder terms, using the Cauchy–Schwarz inequality and the Young inequality, we haveIt follows from (52)–(61) thatDenoteBy (26), we getMultiplying (64) by and integrating over and then replacing by , we getDue to and (43), it is easy to obtain that there exists , such that for all ,By Lemma 2, there are and , such that for all and ,By Lemma 1, there are and , such that for all and ,By equation (17), there is , such that for all ,Denote , ; by (65)–(69), for all and , we havewhich impliesLet with given by (48). Fix and setand then is the solution of problem equation (23) on the bounded domain , where .
Multiplying (23) by and using (72), we find thatConsider the eigenvalue problem.Problem (74) has a family of eigenfunctions with the eigenvalues :such that is an orthonormal basis of . Given , let and be the projection operator.

Lemma 5. Assume that (2)–(7) and (16) hold. Then, for every , and , there exist and and , such that for all , and ,where .

Proof. Let . Applying to , we obtainThen, applying to and taking the inner product of the resulting equation with in , we haveNext, we estimate some terms of (78).Denote ; by (4), we getFor the remainder terms on the right-hand side of (74), we haveBy (30)–(39), we getBy (13), (25), and (89), we getAs , , there exist and such that for all and ,Multiplying (91) by and integrating over and then substituting by , for all and , we haveBy (4), , and , there exist and , such that if and , thenFor the second term on the right-hand side of (92), by (16), we know that there is , such that for all ,For the third and fourth terms on the right-hand side of (93), by Lemma 2, there exist and , such that for all and , there holdsFor the last term on the right-hand side of (93), by Lemma 2, there is , such that for all , it follows thatDenote and . Then, by (92)–(96), for all , , and , we getwhich completes the proof.

5. Random Attractors

In this section, we prove the existence of -pullback attractors for stochastic problem (23).

Lemma 6. Assume that (2)–(7) and (16) hold. Then, the solution of problem (23) is asymptotic compactness in ; that is, for every , and , the sequence has a convergent subsequence in provided and .

Proof. We first let , , and . By Lemma 2, is bounded in ; that is, for every , there exists such for all ,In addition, it follows from Lemma 4 that there exist and , such that for every ,Next, by using Lemma 5, there are , , and , such that for every ,Using (73) and (98), we find that is bounded in the finite-dimensional space , which together with (100) implies that is precompact in .
Note that for . Recalling (73), we find that is precompact in , which along with (99) shows the precompactness of this sequence in . This completes the proof.
The main result of this section is given below.

Theorem 2. Assume that (2)–(7) and (16) hold. Then, the continuous cocycle associated with problem (23) has a unique -pullback attractor in .

Proof. This is an immediate consequence of Proposition 1 and Lemmas 2 and 6.

6. Upper Semicontinuity of Pullback Attractors

In this section, we will use Theorem 1 to consider an upper semicontinuity of random attractors when . To indicate the dependence of solutions on , we, respectively, write the solutions of problem (23) as and , that is, satisfies

When , the random problem (23) reduces to a deterministic one:

By Theorem 2, the deterministic and autonomous system generated by (102) is readily verified to admit a global attractor in .

Theorem 3. Assume that (2)–(7) and (16) hold. Then, the random dynamical system generated by (23) has a unique -pullback attractor in . Moreover, the family of random attractors is upper semicontinuous.

Proof. By Lemma 3 and Theorem 2, has a closed measurable random absorbing set and a unique random attractor .(i)In Lemma 2, we have proved that the system possesses a closed random absorbing set in , which is given by Then, we get which deduces condition (i) in Theorem 1 immediately.(ii)Given , let , where Then, First, by (106), Lemma 4, and the invariance of , we obtain that for every and , there exists such that Second, by (106), the proof of Lemma 5, Lemma 6, and the invariance of , we know that there exists such that for all , the set is precompact in , which together with (107) implies that is precompact in .(iii)Let be a mild solution of (102) with initial data , and , . By (101) and (102), we getTaking the inner product of with in , we getTaking advantage of and Lemma 2, we getBy (7), we getwhich along with (107)-(108) impliesApplying Gronwall inequality to (112) over , we havewhich along with (i), (ii), and Theorem 1 completes the proof [29].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

XY and ZZ completed the main study together. XY wrote the manuscript, and ZZ checked the proofs process and verified the calculation. Moreover, all the authors read and approved the last version of the manuscript.

Acknowledgments

This study was supported by the Natural Science Foundation of China (nos. 12161071 and 11961059).