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Asymptotic behavior of stochastic p-Laplacian-type equation with multiplicative noise

Wenqiang Zhao

Author Affiliations

School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

Boundary Value Problems 2012, 2012:61  doi:10.1186/1687-2770-2012-61

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/61


Received:28 December 2011
Accepted:13 April 2012
Published:22 June 2012

© 2012 Zhao; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The unique existence of solutions to stochastic p-Laplacian-type equation with forced term satisfying some growth and dissipative conditions is established for the initial value in L 2 ( D ) . The generation of a continuous random dynamical system and the existence of a random attractor for stochastic p-Laplacian-type equation driven by multiplicative noise are obtained. Furthermore, we obtain a random attractor consisting of a single point and thus the system possesses a unique stationary solution.

MSC: 60H15, 35B40, 35B41.

Keywords:
random dynamical systems; stochastic p-Laplacian-type equation; random attractors

1 Introduction

The purpose of this paper is to investigate the long-time behavior of solutions to stochastic p-Laplacian-type equation with multiplicative noise, which reads

(1.1)

(1.2)

(1.3)

where Φ p ( s ) = | s | p 2 s , p 2 ; D is an open and bounded subset of R n with regular boundary ∂D; Δ is the Laplacian with regard to the variable x D ; b is a positive constant; u ( t ) = u ( x , t ) a real-valued variable of x D , t 0 ; W ( t ) is mutually independent two-sided real-valued Wiener process defined on a complete probability space ( Ω , F , P ) , where

Ω = { ω C ( R , R ) : ω ( 0 ) = 0 }

and F is the Borel σ-algebra induced by the compact-open topology of Ω, and P is the corresponding Wiener measure on ( Ω , F ) . Then we can identify W ( t ) with ω ( t )

W ( t ) = ω ( t ) , t R .

It is known that the random attractor, which characterizes the long-time behavior of random dynamical systems (RDS) perfectly, was first introduced by [6,13] as a generalization of a global attractor for deterministic PDE. The existences of the random attractor for RDS have been richly developed by many authors for all kinds of SPDEs, see [2,5,6,9,10,15-18,21-25] and references therein.

In deterministic case, there is a large number of works about the p-Laplacian-type equation. Temam [14] obtained the global attractor for (1.1) with exterior forcing term g ( x , u ) = k u , a simple case. In recent years, Yang et al. [19,20] considered the global attractor for a general p-Laplacian-type equation defined both on unbounded domain and bounded domain, respectively. The uniform attractor was also investigated by Chen and Zhong [3] in nonautonomous case. In random case, Zhao [23] obtained random attractors for the p-Laplacian-type equation driven by additive noise.

In this paper, we consider the existence of a random attractor for (1.1)-(1.3) with exterior forcing term g ( x , u ) satisfying some growth conditions. The multiplicative noise c u d W ( t ) characterizes, to some extent, some of the minimal fluctuations among environment or a man-made complex system, which we should take into consideration in order to model perfectly the concrete problem.

One difficulty in our discussions is to estimate the solution operator in the stronger norm space V, where V H V is the Gelfand triple, see Section 2. It seems that the methods used in unperturbed case (see [14,19,20]) are completely unavailable because of the leading term Φ p ( u ) with high order differentials and the forcing term g ( x , u ) with p 1 times growth.

We need to develop some techniques to surmount the obstacle, though we also follow the classic approach (based on the compact embedding) widely used in [5,6,17,21-24] and so on. By using the properties of Dirichlet form for the Laplacian, we overcome this obstacle and obtain the estimate of the solution in the Sobolev space V 0 , which is weaker than V. Here some basic results about the Laplacian are used. We refer to [8] to obtain the details on Dirichlet forms for a negative definite and self-adjoint operator. The existence and uniqueness of a continuous RDS are proved by employing the standard in [12].

We give the outline of this paper. In Section 2, we present some preliminaries for the theory of RDS and the results about the Laplacian which are necessary to our discussion. In Section 3, we prove the existence and uniqueness of a continuous RDS which is generated by the solution to stochastic p-Laplacian-type equation with multiplicative noise. In Section 4, we give some estimates for the solution operators in given Hilbert space and then obtain a random attractor for this RDS. In the last part, we show that the system possesses a unique stationary point under a given condition.

2 Preliminaries

In this section, we present some basic notions about RDS, which can be found in [1,4-6]. We also list the Sobolev spaces, some results about the Laplacian and its Dirichlet forms.

The basic notion in RDS is a metric dynamical system (MSD) θ ( Ω , F , P , { θ t } t R ) , which is a probability space ( Ω , F , P ) with a group θ t , t R , of measure preserving transformations of ( Ω , F , P ) . MSD θ is said to be ergodic under P if for any θ-invariant set B F we have either P ( B ) = 0 or P ( B ) = 1 , where the θ-invariant set is in the sense θ t B = B for B F and all t R .

RDS is an object consisting of a MSD and a cocycle over this MSD, where the MSD is used to model random perturbations. Let X be complete and separable metric space with metric d and Borel sigma-algebra B ( X ) , i.e., the smallest σ-algebra on X which contains all open subsets.

Definition 2.1 (1) A continuous RDS on ( X , d ) over a MSD θ is a family of measurable mappings

φ : R + × Ω × X X , ( t , ω , x ) φ ( t , ω ) x

such that for P -a.e. ω Ω , the mappings φ ( t , ω ) satisfy the cocycle property

φ ( 0 , ω ) = i d , φ ( t + s , ω ) = φ ( t , θ s ω ) φ ( s , ω )

for all s , t R + , and the mappings φ ( t , ω ) : x φ ( t , ω ) x are continuous in X for all t R + .

(2) A continuous stochastic flow is a family of measurable mappings S ( t , s ; ω ) : X X , s t , such that for P -a.e. ω Ω ,

for all s r t , and x S ( t , s ; ω ) x are continuous in X for all s t .

(3) A random compact set { K ( ω ) } ω Ω is a family of compact sets indexed by ω such that for every x X the mapping ω d ( x , K ( ω ) ) is measurable with respect to F .

(4) A random set { A ( ω ) } ω Ω is an attracting set if for every deterministic bounded subset B X and P -a.e. ω Ω ,

lim t dist ( φ ( t , θ t ω ) B , A ( ω ) ) = 0 ,

where dist ( , ) is defined by dist ( A , B ) = sup x A inf y B d ( x , y ) .

(5) A random set { A ( ω ) } ω Ω is an absorbing set if for every deterministic bounded subset B X and P -a.e. ω Ω , there exists t B ( ω ) > 0 such that for all t t B ( ω ) ,

φ ( t , θ t ω ) B A ( ω ) ,

where φ ( t , θ t ω ) B = x B φ ( t , θ t ω ) x .

It is obvious that an absorbing set is an attracting set. The attraction in the definition of the attracting set is a form of pathwise convergence. In fact, the attracting set also attracts in the weaker convergence in probability, in the sense, for all ε > 0 and every bounded set B X ,

lim t P ( dist X ( φ ( t , θ t ω ) B , A ( ω ) ) > ε ) = 0 .

Definition 2.2 A random compact set ω A ( ω ) is called to be a random attractor for the RDS φ if { A ( ω ) } ω Ω is an attracting set and φ ( t , ω ) A ( ω ) = A ( θ t ω ) for ω Ω and all t 0 .

Theorem 2.3 (see [4])

Assume that φ ( t , ω ) is a continuous RDS onXover MDSθ. If there exists a compact random absorbing set { K ( ω ) } ω Ω , then φ ( t , ω ) possesses a random attractor { A ( ω ) } ω Ω defined by

A ( ω ) = B B ( X ) s 0 t s φ ( t , θ t ω ) B ¯ ¯ , (2.1)

where B ( X ) denotes all the bounded subsets ofX.

Let L p ( D ) be the p-times integrable functions space on D with norm denoted by p , and V = W 0 2 , p ( D ) with Sobolev equivalent norm (see p.166 of [14])

v V = Δ v p = ( D | Δ v | p d x ) 1 p , v V .

Put the dual V of V by V = W 2 , p ( D ) , where

u W 2 , p ( D ) u = | α | 2 D α f α , f α L p ( D )

and 1 p + 1 p = 1 . Let H = L 2 ( D ) with the usual scalar product and norm { ( , ) , 2 } . Then we have the following Gelfand triple

V H H V ,

or concretely

W 0 2 , p ( D ) L 2 ( D ) W 2 , p p 1 ( D ) ,

where the injections are continuous and each space is dense in the following one.

We know that the Laplacian Δ, which is negative definite and self-adjoint, is the generator (with domain W 0 2 , p ( D ) ) of a strongly continuous semigroup M ( t ) on L p ( D ) which is contractive and positive. Here “contractive” means M ( t ) p 1 and “positive” means M ( t ) u 0 for every 0 u L p ( D ) . The resolvent of generator Δ is denoted by R ( λ , Δ ) λ ρ ( Δ ) , where ρ ( Δ ) is the resolvent set of Δ. By the Lumer-Phillips Theorem in [11], it follows that ( 0 , ) ρ ( Δ ) and for u L p ( D )

λ R ( λ , Δ ) u p u p , λ > 0 .

Moreover, for u D ( Δ ) R ( λ , Δ ) u D ( Δ ) and Δ R ( λ , Δ ) u = R ( λ , Δ ) Δ u , where D ( Δ ) is the domain of Δ.

Since Δ is negative definite and self-adjoint, then Δ is associated with the Dirichlet forms E by

E ( u , v ) = ( Δ u , Δ v ) , u , v H 0 1 ( D ) H 2 ( D ) . (2.2)

E is unique determined by Δ. For u , v H 0 1 ( D ) H 2 ( D ) , we define a new inner product by

E ( λ ) ( u , v ) = λ ( u λ R ( λ , Δ ) u , v ) , λ > 0 , (2.3)

where R ( λ , Δ ) is the resolvent of Δ. Then it follows from Ref. [8] that E ( λ ) ( u , v ) as λ , and

lim λ E ( λ ) ( u , v ) = ε ( u , v ) (2.4)

for u , v H 0 1 ( D ) H 2 ( D ) .

3 Existence and uniqueness of RDS

In this section, we show the existence and uniqueness of a continuous RDS for the following stochastic p-Laplacian-type equation with multiplicative noise,

(3.1)

(3.2)

(3.3)

where Φ p ( s ) = | s | p 2 s , p 2 . To study System (3.1)-(3.3), we assume that the nonlinearity g ( x , u ) defined in D × R satisfies the following conditions:

(3.4)

(3.5)

(3.6)

where 2 q p < .

For ω Ω , we define a nonlinear operator A on V by

A ( u ( t ) , ω ) = Δ Φ ( Δ u ( t ) ) + g ( x , u ( t ) ) f ( x ) , u ( t ) V , x D . (3.7)

Then (3.1) reads

d u ( t ) d t + A ( u ( t ) , ω ) = b u ( t ) d W ( t ) . (3.8)

Since p q , by our assumption (3.4)-(3.6) and f V , it is easy to check that for given ω Ω , the operator A : u A ( u , ω ) mapping W 2 , p ( D ) into W 2 , p ( D ) is well defined, where p = p p 1 .

Let ( Ω , F , P ) be the probability space as in the introduction. Define the Wiener shift by

θ t ω ( s ) = ω ( s + t ) ω ( t ) , ω Ω , t , s R .

Then θ = ( Ω , F , P , { θ t } t R ) is an ergodic MDS.

In order to obtain the existence of a continuous RDS, it is necessary to translate (3.1)-(3.3) into a deterministic system parameterized by ω. To this end, we consider the Ornstein-Uhlenbeck process. Put

t z ( θ t ω ) : = 0 e τ ( θ t ω ) ( τ ) d τ , t R ,

which solves the Itô differential equation

d z ( θ t ω ) + z ( θ t ω ) d t = d W ( t ) , z ( ) = 0 ,

where the Ornstein-Uhlenbeck constant equals to 1.

Note that z ( θ t ω ) is a Gaussian process with mathematical expectation E [ z ( θ t ω ) ] 0 and variance σ 2 ( t ) = 1 2 , see [7], whereas lim t 1 t 0 t z ( θ τ ω ) d τ = 0 . Furthermore, from [2,15,18], the random variable | z ( ω ) | is continuous in t for P -a.e. ω Ω and grows sublinearly, i.e., lim t ± | z ( θ t ω ) | | t | = 0 .

We now translate (3.1) by one classical change of variables

v ( t ) = e b z ( θ t ω ) u ( t ) . (3.9)

Then we have

d u ( t ) = e b z ( θ t ω ) d v ( t ) b e b z ( θ t ω ) z ( θ t ω ) v ( t ) d t + b u ( t ) d W ( t ) .

Then, formally, the variable v ( t ) satisfies the following equations parameterized by ω Ω but without white noise:

(3.10)

(3.11)

(3.12)

where g ( x , u ) satisfies (3.4)-(3.6) and f is given in V , 2 q p < .

For convenience, we put

A ¯ ( v ( t ) , ω ) = e b ( p 2 ) z ( θ t ω ) Δ Φ ( Δ v ) + e b z ( θ t ω ) g ( x , e b z ( θ t ω ) v ) e b z ( θ t ω ) f ( x ) b z ( θ t ω ) v . (3.13)

Then we have

d v ( t ) d t + A ¯ ( v ( t ) , ω ) = 0 . (3.14)

Note that System (3.1)-(3.3) and System (3.10)-(3.12) are equivalent by (3.9). Let u ( t , ω ; s , u 0 ) and v ( t , ω ; s , v 0 ) be the solution of System (3.1)-(3.3) and System (3.10)-(3.12) respectively. It is easy to check that if System (3.1)-(3.3) possess a unique solution in V for all initial values in H then System (3.10)-(3.12) possess a unique solution in V for the same initial value in H. Moreover, if the mapping v 0 v ( t , ω ; s , v 0 ) is continuous in H for the initial value in H, then the mapping u 0 u ( t , ω ; s , u 0 ) is also continuous in H, vice verse.

We now show the existence and uniqueness of solution to System (3.1)-(3.6).

Theorem 3.1Assume thatgsatisfies (3.4)-(3.6) andfis given in V , 2 q p < . Then for all u 0 H with u 0 = u ( s ) , System (3.1)-(3.3) has a unique solution

u ( t , ω ; s , u 0 ) L l o c p ( [ s , ) , V ) C ( [ s , ) , H )

for all t s and P -a.e. ω Ω . Furthermore, the mapping u 0 u ( t , ω ; s , u 0 ) fromHintoHis continuous for all t s .

Proof We first show that for every u 0 H there exists a unique solution u ( t , ω ; s , u 0 ) L l o c p ( [ s , ) , V ) . By Theorem 4.2.4 and Exercise 4.1.2 in [12], it suffices to show that for every fixed ω Ω A ( u ( t ) , ω ) possesses Hemi-continuity, Monotonicity, Coercivity, and Bounded-ness properties (for the definitions of these notions please refer to p.56 of [12]). But the proofs are an analogy of the corresponding works in [23]. So we omit them here.

We then show that the solution is in C ( [ s , T ] , H ) . By our assumptions that p q and f V , we can check that A ¯ ( v ( t ) , ω ) maps L p ( [ s , T ] , V ) to L p ( [ s , T ] , V ) for ω Ω . Thus if v L p ( [ s , T ] , V ) , then (3.14) implies that d v d t L p ( [ s , T ] , V ) . Now by the general fact (see p.164 of [14]) it follows that v is almost everywhere equal to a function belonging to C ( [ s , T ] , H ) . Hence by the transformation (3.9) and the continuous property of Ornstein-Uhlenbeck process, u ( t , ω ; s , u 0 ) is almost everywhere equal to a function belonging to C ( [ s , T ] , H ) .

We finally prove the continuity of the mapping u 0 u ( t , ω ; s , u 0 ) from H into H. It suffices to prove that the mapping v 0 v ( t , ω ; s , v 0 ) is continuous from H into H.

Let v 1 , v 2 be two different initial values at initial value time s, and corresponding solutions be denoted by v 1 ( t , ω ; s , v 1 ) and v 2 ( t , ω ; s , v 2 ) respectively. Then it follows from (3.14) that

d d t ( v 1 ( t ) v 2 ( t ) ) + A ¯ ( v 1 ( t ) , ω ) A ¯ ( v 2 ( t ) , ω ) = 0 , (3.15)

where A ¯ ( v ( t ) , ω ) is defined in (3.13). Note that

Because the function u p 1 is increasing for u 0 and p 2 , the last inequality in the above proof is correct. Then by a simple computation we find that for fixed ω Ω ,

( A ¯ ( v 1 ( t ) , ω ) A ¯ ( v 2 ( t ) , ω ) , v 1 ( t ) v 2 ( t ) ) ( k 3 b z ( θ t ω ) ) v 1 ( t ) v 2 ( t ) 2 2 , (3.16)

where k 3 is in (3.6). Hence, multiplying (3.15) by v 1 ( t ) v 2 ( t ) , integrating over D, and using (3.16), we get that

d d t v 1 ( t ) v 2 ( t ) 2 2 + 2 ( k 3 b z ( θ t ω ) ) v 1 ( t ) v 2 ( t ) 2 2 0 . (3.17)

Using Gronwall’s lemma to (3.17) from s to t, it yields that

v 1 ( t ) v 2 ( t ) 2 2 e s t 2 ( k 3 + b z ( θ τ ω ) ) d τ v 1 v 2 2 2 . (3.18)

Then, the continuity of the mapping v 0 v ( t , ω ; s , v 0 ) from H into H is followed from the contraction property (3.18). This finishes the total proofs of Theorem 3.1. □

We now define

S ( t , s ; ω ) u 0 = u ( t , ω ; s , u 0 ) , t s R , (3.19)

with u 0 = u ( s ) . By the uniqueness part of the solution in Theorem 3.1, we immediately get that S ( t , s ; ω ) is a stochastic flow; that is, for every u 0 H and t r s R

(3.20)

(3.21)

Hence if we define

ψ ( t , ω ) u 0 = S ( t , 0 ; ω ) u 0 = u ( t , ω ; 0 , u 0 )

with u 0 = u ( 0 ) , then by Theorem 3.1 ψ is a continuous RDS associated with System (3.1)-(3.3).

We define

φ ( t , ω ) v 0 = v ( t , ω ; 0 , v 0 ) = e b z ( θ t ω ) ψ ( t , ω ) u 0 = e b z ( θ t ω ) u ( t , ω ; 0 , u 0 ) .

Then φ is a continuous RDS associated with System (3.10)-(3.12), with the following fact

φ ( t , θ t ω ) v 0 = v ( 0 , ω ; t , v 0 ) for all t 0 . (3.22)

That is to say, φ ( t , θ t ω ) v 0 can be interpreted as the position of the trajectory at time 0, which was in v 0 at time −t (see [5]).

It is easy to check that ψ possesses a random attractor provided that φ possesses a random attractor. Hence in the following we only concentrate on the RDS φ.

4 Existence of compact random attractor for RDS

In this section, we will compute some estimates in space H = L 2 ( D ) and V 0 = H 0 1 ( D ) . Note that in the following ω Ω ; the results will hold for P -a.e. ω Ω and the generic constants c or c i , i = 1 , 2 , are independent of λ > 0 in the context, where λ ρ ( Δ ) .

Lemma 4.1Suppose thatgsatisfies (3.4)-(3.6) andfis given in V . Then there exist random radii r 1 ( ω ) , r 2 ( ω ) > 0 , such that for all ϱ > 0 there exists s = s ( ω , ϱ ) 1 such that for all s s ( ω , ϱ ) and all v 0 H with v 0 2 ϱ , the following inequalities hold for P -a.e. ω Ω ,

where v ( t , ω ; s , v 0 ) is the solution to Equation (3.10) with v 0 = v ( s ) .

Proof For simplicity, we abbreviate v ( t ) = : v ( t , ω ; s , v 0 ) for t s with v 0 = v ( s ) . Multiplying both sides of (3.10) by v ( t ) and then integrating over D, we obtain that

1 2 d d t v 2 2 + e b ( p 2 ) z ( θ t ω ) ( Δ Φ ( Δ v ) , v ) + e b z ( θ t ω ) D g ( x , e b z ( θ t ω ) v ) v d x = e b z ( θ t ω ) ( f , v ) + b z ( θ t ω ) v 2 2 , (4.1)

where

(4.2)

(4.3)

(4.4)

Then by (4.1)-(4.4), we have

(4.5)

Since q 2 , then by using Sobolev’s embedding inequality and inverse Young’s inequality we see that

2 k 1 e b ( q 2 ) z ( θ t ω ) v q q 2 c k 1 e b ( q 2 ) z ( θ t ω ) v 2 q 2 v 2 2 2 c e 2 b z ( θ t ω ) . (4.6)

Then it follows from (4.5) and (4.6) that

d d t v 2 2 ( 2 + 2 b z ( θ t ω ) ) v 2 2 + c e 2 b z ( θ t ω ) . (4.7)

By employing Gronwall’s lemma over interval [ s , t ] with t [ 1 , 0 ] , we find that

v ( t ) 2 2 e s t ( 2 + 2 b z ( θ τ ω ) ) d τ v ( s ) 2 2 + c s t e 2 b z ( θ τ ω ) e τ t ( 2 + 2 b z ( θ σ ω ) ) d σ d τ = e t 0 ( 2 2 b z ( θ τ ω ) ) d τ ( e s 0 ( 2 + 2 b z ( θ τ ω ) ) d τ v ( s ) 2 2 + c 0 e 2 b z ( θ τ ω ) + τ 0 2 b z ( θ σ ω ) d σ + 2 τ d τ ) e 1 0 ( 2 + 2 b | z ( θ τ ω ) | ) d τ ( e s 0 ( 2 + 2 b z ( θ τ ω ) ) d τ ϱ 2 + c 0 e 2 b z ( θ τ ω ) + τ 0 2 b z ( θ σ ω ) d σ + 2 τ d τ ) (4.8)

for v ( s ) 2 2 ϱ 2 . By the properties of the Ornstein-Uhlenbeck process, we deduce that

lim s e s 0 ( 2 + 2 b z ( θ τ ω ) ) d τ ϱ 2 = 0 (4.9)

and

0 e 2 b z ( θ τ ω ) + τ 0 2 b z ( θ σ ω ) d σ + 2 τ d τ < + . (4.10)

Hence, given every fixed ϱ > 0 and v ( s ) 2 2 ϱ 2 , we can choose s ( ω , ϱ ) 1 , depending only on ω and ϱ, such that for all s s ( ω , ϱ ) and t [ 1 , 0 ] ,

v ( t , ω ; s , v 0 ) 2 2 e 1 0 ( 2 + 2 b | z ( θ τ ω ) | ) d τ ( 1 + c 0 e 2 b z ( θ τ ω ) + c τ 0 2 b z ( θ σ ω ) d σ + 2 τ d τ ) , (4.11)

which gives an expression for r 1 2 ( ω ) . Replacing t by τ in (4.5) and integrating for τ over intervals [ 1 , 0 ] , then using (4.11) it yields that for all s s ( ω , ϱ ) ,

(4.12)

Then we have

1 0 ( Δ v ( τ , ω ; s , v 0 ) p p + v ( τ , ω ; s , v 0 ) q q ) d τ m ( ω ) ( 2 b r 1 2 ( ω ) 1 0 z ( θ τ ω ) d τ + c 1 0 e 2 b z ( θ τ ω ) d τ + r 1 2 ( ω ) ) , (4.13)

where

m ( ω ) 1 = min { min 1 t 0 { e b ( p 2 ) z ( θ τ ω ) } , min 1 t 0 { 2 k 1 e b ( q 2 ) z ( θ τ ω ) } } .

Thus the right-hand side of (4.13) gives an expression for r 2 2 ( ω ) . □

In the following, we shall obtain the regularity of the solution to stochastic p-Laplacian-type equation. This is the most challenging part in our discussion. Because of the nonlinearity of driven Δ Φ ( Δ u ) and function g ( x , u ) in Equation (3.10), it seems difficult to derive the V-norm estimate as in [14], where the author only deals with a linear case, i.e., g ( x , u ) = k u . So we relax to estimate the solution in a weaker Sobolev V 0 = H 0 1 ( D ) with equivalent norms denoted by v 2 for v V 0 . Here, just as stated in the introduction, we use the properties of Dirichlet forms for the Laplacian Δ.

Lemma 4.2Suppose thatgsatisfies (3.4)-(3.6) andfis given in V . Then there exists a random radius r 3 ( ω ) > 0 , such that for all ϱ > 0 there exists s = s ( ω , ϱ ) 1 such that for all s s ( ω , ϱ ) and all v 0 H with v 0 2 ϱ , the following inequality holds for P -a.e. ω Ω

v ( t , ω ; s , v 0 ) 2 2 r 3 2 ( ω ) for all t [ 1 , 0 ] ,

where v ( t , ω ; s , v 0 ) is the solution to (3.10) with v 0 = v ( s ) .

Proof Taking the inner product of (3.10) with λ Δ R ( λ , Δ ) v where λ > 0 and v V , we get that

D v t λ Δ R ( λ , Δ ) v d x = e b ( p 2 ) z ( θ t ω ) D Δ Φ ( Δ v ) λ Δ R ( λ , Δ ) v d x + e b z ( θ t ω ) D g ( x , e b z ( θ t ω ) v ) λ Δ R ( λ , Δ ) v d x e b z ( θ t ω ) D f ( x ) λ Δ R ( λ , Δ ) v d x b z ( θ t ω ) D v λ Δ R ( λ , Δ ) v d x . (4.14)

By the semigroup theory (see [14]) we have

Δ R ( λ , Δ ) v = R ( λ , Δ ) Δ v = λ R ( λ , Δ ) v v (4.15)

for v D ( Δ ) , the domain of Laplacian Δ. We now estimate all terms on the right-hand side of (4.14). Employing (4.15) and integrating by parts, it yields that

D Δ Φ ( Δ v ) λ Δ R ( λ , Δ ) v = λ D Δ Φ ( Δ v ) ( λ R ( λ , Δ ) v v ) d x = λ D Δ ( | Δ v | p 2 Δ v ) v d x + λ D Δ ( | Δ v | p 2 Δ v ) λ R ( λ , Δ ) v d x = λ Δ v p p + λ D ( | Δ v | p 2 Δ v ) λ Δ R ( λ , Δ ) v d x λ Δ u p p + λ D | Δ u | p 1 | λ R ( λ , Δ ) Δ v | d x λ Δ v p p + λ Δ v p p 1 λ R ( λ , Δ ) Δ v p λ Δ v p p + λ Δ v p p = 0 , (4.16)

where we use the contraction property of λ R ( λ , Δ ) on L p ( D ) , i.e.,

λ R ( λ , Δ ) Δ v p Δ v p (4.17)

for Δ v L p ( D ) and every λ > 0 . By our assumption (3.5), along with (4.17) for q, the second term on the right-hand side of (4.14) is estimated as

e b z ( θ t ω ) D g ( x , e b z ( θ t ω ) v ) λ Δ R ( λ , Δ ) v d x e b z ( θ t ω ) D | g ( x , e b z ( θ t ω ) v ) | | λ R ( λ , Δ ) Δ v | d x e b z ( θ t ω ) D ( k 2 e b ( q 1 ) z ( θ t ω ) | v | q 1 + ϕ 2 ( x ) ) | λ R ( λ , Δ ) Δ v | d x k 2 e b ( q 2 ) z ( θ t ω ) v q q 1 λ R ( λ , Δ ) Δ v q + e b z ( θ t ω ) ϕ 2 q λ R ( λ , Δ ) Δ v q k 2 e b ( q 2 ) z ( θ t ω ) v q q 1 Δ v q + e b z ( θ t ω ) ϕ 2 q Δ v q k 2 e b ( q 2 ) z ( θ t ω ) v q q + k 2 e b ( q 2 ) z ( θ t ω ) Δ v q q + e b z ( θ t ω ) Δ v q q + e b z ( θ t ω ) ϕ 2 q q , (4.18)

where we employ Young’s inequality a b a r + b r r 1 for r > 1 twice. But, by Sobolev’s inequality and Young’s inequality, it yields that

k 2 e b ( q 2 ) z ( θ t ω ) Δ v q q c e b ( q 2 ) z ( θ t ω ) Δ v p q = e b q ( p 2 ) p z ( θ t ω ) Δ v p q . c e 2 b ( q p ) p z ( θ t ω ) e b ( p 2 ) z ( θ t ω ) Δ v p p + c e 2 b z ( θ t ω ) (4.19)

and by (4.19) we have

e b z ( θ t ω ) Δ v q q = 1 k 2 e b ( q 1 ) z ( θ t ω ) k 2 e b ( q 2 ) z ( θ t ω ) Δ v q q 1 k 2 e b ( q 1 ) z ( θ t ω ) ( e b ( p 2 ) z ( θ t ω ) Δ v p p + c e 2 b z ( θ t ω ) ) 1 k 2 e b ( p q 1 ) z ( θ t ω ) Δ v p p + c e b ( q + 1 ) z ( θ t ω ) . (4.20)

Then by (4.18)-(4.20), there exist positive constants c such that

(4.21)

For the third term on the right-hand side of (4.14), by (4.17) we see that

e b z ( θ t ω ) D f ( x ) λ Δ R ( λ , Δ ) v d x e b z ( θ t ω ) f p λ R ( λ , Δ ) Δ v p e b z ( θ t ω ) f p Δ v p e b ( p 2 ) z ( θ t ω ) Δ v p p + e 2 b z ( θ t ω ) f p p . (4.22)

On the other hand, by (4.15) and the Dirichlet forms E (2.3), we have

D v t λ Δ R ( λ , Δ ) v d x = E ( λ ) ( v , v t ) , D v λ Δ R ( λ , Δ ) v d x = E ( λ ) ( v , v ) . (4.23)

Then it follows from (4.14), (4.16) and (4.21)-(4.23) that

E ( λ ) ( v , v t ) p 1 ( θ t ω ) Δ v p p + p 2 ( θ t ω ) v q q + p 3 ( θ t ω ) + b z ( θ t ω ) E ( λ ) ( v , v ) , (4.24)

where

and c is a positive constant independent of λ. So taking limit on both sides of (4.24) for λ and associating with (2.2) and (2.4), we deduce that

1 2 d d t v ( t ) 2 2 p 1 ( θ t ω ) Δ v p p + p 2 ( θ t ω ) v q q + b z ( θ t ω ) v ( t ) 2 2 + p 3 ( θ t ω ) . (4.25)

Replacing t by τ in (4.25) and integrating τ from s to t ( 1 s t 0 ), it yields that

v ( t ) 2 2 2 s t p 1 ( θ τ ω ) Δ v ( τ ) p p d τ + 2 s t p 2 ( θ τ ω ) v ( τ ) q q d τ + 2 s t p 3 ( θ τ ω ) d τ + 2 b s t z ( θ τ ω ) v ( τ ) 2 2 d τ + v ( s ) 2 2 2 1 0 p 1 ( θ τ ω ) Δ v ( τ ) p p d τ + 2 1 0 p 2 ( θ τ ω ) v ( τ ) q q d τ + 2 1 0 p 3 ( θ τ ω ) d τ + 2 b 1 0 z ( θ τ ω ) v ( τ ) 2 2 d τ + v ( s ) 2 2 . (4.26)

Put

M ( ω ) = max { max 1 τ 0 { p 1 ( θ τ ω ) } , max 1 τ 0 { p 2 ( θ τ ω ) } , max 1 τ 0 { p 3 ( θ τ ω ) } , max 1 τ 0 { z ( θ τ ω ) } } .

Then by Lemma 4.1, (4.26) reads

v ( t ) 2 2 2 M ( ω ) r 2 2 ( ω ) + 2 M ( ω ) + 2 b M ( ω ) 1 0 v ( τ ) 2 2 d τ + v ( s ) 2 2 . (4.27)

Integrating (4.27) for s over intervals [ 1 , 0 ] , we have

v ( t ) 2 2 2 M ( ω ) r 2 2 ( ω ) + 2 M ( ω ) + ( 2 b M ( ω ) + 1 ) 1 0 v ( τ ) 2 2 d τ (4.28)

for all t [ 1 , 0 ] . By Poincare’s inequality and Young’s inequality, there exists positive constant c such that

v ( s ) 2 2 c Δ v ( s ) 2 2 Δ v ( s ) p p + c . (4.29)

Hence by using Lemma 4.1 again, along with (4.29), it follows from (4.28) that

v ( t ) 2 2 2 M ( ω ) r 2 2 ( ω ) + 2 M ( ω ) + ( 2 b M ( ω ) + 1 ) r 2 ( ω ) + c ,

with t [ 1 , 0 ] , which gives an expression for r 3 2 ( ω ) . This completes the proof. □

By Theorem 2.3 and Lemma 4.2, we have obtained our main result in this section.

Theorem 4.3Assume thatgsatisfies (3.4)-(3.6) andfis given in V . Then the RDS φ ( t , ω ) generated by System (3.10)-(3.12) possesses a random attractor { A ( ω ) } ω Ω defined by

A ( ω ) = B B ( H ) s 0 t s φ ( t , θ t ω ) B ¯ ¯ ,

where B ( H ) denotes all the bounded subsets ofHand the closure is theH-norm.

5 The single point attractor

In this section, we consider a special case, that is, k 3 > 0 in (3.6), in which case we find that the random attractor is just composed of a single point. This shows that System (3.10)-(3.12) possesses an unique stationary solution for every given initial value in the space H. We begin with a lemma.

Lemma 5.1Assume thatgsatisfies (3.4)-(3.6) andfis given in V , k 3 > 0 . Then for s 1 s 2 t and v ( s 1 ) , v ( s 2 ) H , there exists a positive constant k < k 3 such that

In particular, for each fixed t R and ω Ω there exists a single point ς t ( ω ) inHsuch that

lim s v ( t , ω ; s , v ( s ) ) = ς t ( ω ) , (5.1)

for every v ( s ) belonging to the bounded subsetBofH. Furthermore, the convergence in (5.1) is uniform with respect to all v ( s ) B .

Proof Let v ( t , ω ; s i , v ( s i ) ) be the solutions to (3.10) with initial values v ( s i ) H , i = 1 , 2 . Then we can deduce from (3.14) that

d d t ( v ( t , ω ; s 1 , v ( s 1 ) ) v ( t , ω ; s 2 , v ( s 2 ) ) ) + A ¯ ( v ( t , ω ; s 1 , v ( s 1 ) ) , ω ) A ¯ ( v ( t , ω ; s 2 , v ( s 2 ) ) , ω ) = 0 . (5.2)

Multiplying (5.2) by v ( t , ω ; s 1 , v ( s 1 ) ) v ( t , ω ; s 2 , v ( s 2 ) ) , integrating over D and using (3.16), we find that

d d t v ( t , ω ; s 1 , v ( s 1 ) ) v ( t , ω ; s 2 , v ( s 2 ) ) 2 2 + ( k 3 b z ( θ t ω ) ) v ( t , ω ; s 1 , v ( s 1 ) ) v ( t , ω ; s 2 , u ( s 2 ) ) 2 2 0 . (5.3)

Now, applying Gronwall’s lemma to (5.3) from s 2 to t, it yields that

v ( t , ω ; s 1 , v ( s 1 ) ) v ( t , ω ; s 2 , v ( s 2 ) ) 2 2 v ( s 2 , ω ; s 1 , v ( s 1 ) ) v ( s 2 ) 2 2 e s 2 t ( k 3 b z ( θ τ ω ) ) d τ 2 ( v ( s 2 , ω ; s 1 , v ( s 1 ) ) 2 2 + v ( s 2 ) 2 2 ) e s 2 t ( k 3 b z ( θ τ ω ) ) d τ = 2 e 0 t ( k 3 b z ( θ τ ω ) ) d τ ( v ( s 2 , ω ; s 1 , v ( s 1 ) ) 2 2 + v ( s 2 ) 2 2 ) e s 2 0 ( k 3 b z ( θ τ ω ) ) d τ . (5.4)

We then estimate v ( s 2 , ω ; s 1 , v ( s 1 ) ) 2 2 . By (4.5) we have

d d t v ( t , ω ; s 1 , v ( s 1 ) ) 2 2 + 2 k 1 e b ( q 2 ) z ( θ t ω ) v ( t , ω ; s 1 , v ( s 1 ) ) q q 2 b z ( θ t ω ) v ( t , ω ; s 1 , v ( s 1 ) ) 2 2 + c e 2 b z ( θ t ω ) . (5.5)

By the Sobolev’s embedding inequality and the inverse Young’s inequality, we can choose 0 < k < k 3 such that

2 k 1 e b ( q 2 ) z ( θ t ω ) v ( t , ω ; s 1 , v ( s 1 ) ) q q 2 c k 1 e b ( q 2 ) z ( θ t ω ) v ( t , ω ; s 1 , v ( s 1 ) ) 2 q k v ( t , ω ; s 1 , v ( s 1 ) ) 2 2 c e 2 b z ( θ t ω ) . (5.6)

So by (5.5) and (5.6) we get that

d d t v ( t , ω ; s 1 , v ( s 1 ) ) 2 2 ( k + 2 b z ( θ t ω ) ) v ( t , ω ; s 1 , v ( s 1 ) ) 2 2 + c e 2 b z ( θ t ω ) . (5.7)

Using Gronwall’s lemma to (5.7) from s 1 to s 2 with s 1 s 2 0 , we get that

(5.8)

Similar to the argument of (4.10), we know that the integral in the last term on the right-hand side of (5.8) is convergent. Hence, it follows from (5.8) and (5.4) that for every fixed t R ,

(5.9)

So for every bounded subset B of H and v ( s 1 ) , v ( s 2 ) B , it follows from (5.9) that

v ( t , ω ; s 1 , v ( s 1 ) ) v ( t , ω ; s 2 , v ( s 2 ) ) 2 2 0 as s 1 , s 2 , (5.10)

since by the properties of the Ornstein-Uhlenbeck process, we have

lim s 2 v ( s 2 ) 2 2 e s 2 0 ( k 3 + b z ( θ τ ω ) ) d τ = 0 (5.11)

and

lim s 1 v ( s 1 ) 2 2 e s 1 0 ( k + b z ( θ τ ω ) ) d τ = 0 (5.12)

and

lim s 2 ( 0 e 2 b z ( θ τ ω ) + τ 0 ( k + 2 b z ( θ σ ω ) ) d σ d τ ) e s 2 0 ( k k 3 b z ( θ τ ω ) ) d τ = 0 . (5.13)

Moreover, the convergence in (5.11)-(5.13) is uniform with respect to v ( s 1 ) , v ( s 2 ) belonging to every bounded subset of H. Then (5.10) implies that for fixed t R , v ( t , ω ; s , v ( s ) ) is a Cauchy sequence in H with respect to s . Therefore, by the completeness of H, for every fixed t R and ω Ω , v ( t , ω ; s , v ( s ) ) has a limit in H denoted by ς t ( ω ) , i.e.,

lim s v ( t , ω ; s , v ( s ) ) = ς t ( ω ) .

 □

Theorem 5.2Assume thatgsatisfies (3.4)-(3.6) andfis given in V , k 3 > 0 . Then the RDS φ ( t , ω ) generated by the solution to (3.10)-(3.12) possesses a single point attractor { A ( ω ) } ω Ω , i.e., there exists a single point ς 0 ( ω ) inHsuch that

A ( ω ) = { ς 0 ( ω ) } .

Proof Put

S ¯ ( t , s ; ω ) = v ( t , ω ; s , v ( s ) ) .

Then S ¯ ( t , s ; ω ) is a stochastic flow associated with System (3.10)-(3.12) and the RDS φ ( t , ω ) = S ¯ ( t , 0 ; ω ) . By Lemma 5.1 we define

ς 0 ( ω ) = lim s S ¯ ( 0 , s ; ω ) v 0 ,

where v 0 = v ( s ) . Then we need to prove that { A ( ω ) } ω Ω = { ς 0 ( ω ) } ω Ω is a compact attractor. It is obvious that { ς 0 ( ω ) } ω Ω is a compact random set. Hence by Definition 2.2 it suffices to prove the invariance and attracting property for { ς 0 ( ω ) } ω Ω . Since by the continuity of φ ( t , ω ) and the flow properties of S ¯ ( t , s ; ω ) , we have

φ ( t , ω ) ς 0 ( ω ) = φ ( t , ω ) lim s S ¯ ( 0 , s ; ω ) v 0 = lim s φ ( t , ω ) S ¯ ( 0 , s ; ω ) v 0 = lim s S ¯ ( t , 0 ; ω ) S ¯ ( 0 , s ; ω ) v 0 = lim s S ¯ ( t , s ; ω ) v 0 = lim s S ¯ ( t s , 0 ; θ s ω ) v 0 = lim s S ¯ ( 0 , s t ; θ t ω ) v 0 = ς 0 ( θ t ω ) .

That is to say, φ ( t , ω ) A ( ω ) = A ( θ t ω ) . On the other hand, by the uniform convergence of (5.1), it follows from (3.22) that for every bounded subset B H ,

dist ( φ ( t , θ t ω ) B , A ( ω ) ) = sup v 0 B φ ( t , θ t ω ) v 0 ς 0 ( ω ) 2 = sup v 0 B S ¯ ( 0 , t , ω ) v 0 ς 0 ( ω ) 2 0

as t + . This shows that { A ( ω ) } ω Ω is an attracting set, and thus we complete the proof. □

Competing interests

The author declares that he has no competing interests.

Author’s contributions

ZW carried out all studies in this article.

Acknowledgements

The author is indebted to the referee for giving some valuable suggestions that improved the presentations of this article. This work was supported by the China NSF Grant (no. 10871217), the Science and Technology Funds of Chongqing Educational Commission (no. KJ120703), the Fundamental Funds of the Central Universities (no. XDJK2009C100) and the Doctor Funds of Southwest University (no. SWU111068).

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