Abstract
The unique existence of solutions to stochastic pLaplaciantype equation with forced term satisfying some growth and dissipative conditions is established for the initial value in . The generation of a continuous random dynamical system and the existence of a random attractor for stochastic pLaplaciantype 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 pLaplaciantype equation; random attractors1 Introduction
The purpose of this paper is to investigate the longtime behavior of solutions to stochastic pLaplaciantype equation with multiplicative noise, which reads
where , ; D is an open and bounded subset of with regular boundary ∂D; Δ is the Laplacian with regard to the variable ; b is a positive constant; a realvalued variable of , ; is mutually independent twosided realvalued Wiener process defined on a complete probability space , where
and is the Borel σalgebra induced by the compactopen topology of Ω, and is the corresponding Wiener measure on . Then we can identify with
It is known that the random attractor, which characterizes the longtime 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 [2,5,6,9,10,1518,2125] and references therein.
In deterministic case, there is a large number of works about the pLaplaciantype equation. Temam [14] obtained the global attractor for (1.1) with exterior forcing term , a simple case. In recent years, Yang et al. [19,20] considered the global attractor for a general pLaplaciantype 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 pLaplaciantype 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 satisfying some growth conditions. The multiplicative noise characterizes, to some extent, some of the minimal fluctuations among environment or a manmade 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 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 with high order differentials and the forcing term with 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 [5,6,17,2124] 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 , 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 selfadjoint 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 pLaplaciantype 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 [1,46]. 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) , which is a probability space with a group , of measure preserving transformations of . MSD θ is said to be ergodic under if for any θinvariant set we have either or , where the θinvariant set is in the sense for and all .
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 sigmaalgebra , i.e., the smallest σalgebra on X which contains all open subsets.
Definition 2.1 (1) A continuous RDS on over a MSD θ is a family of measurable mappings
such that for a.e. , the mappings satisfy the cocycle property
for all , and the mappings are continuous in X for all .
(2) A continuous stochastic flow is a family of measurable mappings , , such that for a.e. ,
for all , and are continuous in X for all .
(3) A random compact set is a family of compact sets indexed by ω such that for every the mapping is measurable with respect to .
(4) A random set is an attracting set if for every deterministic bounded subset and a.e. ,
(5) A random set is an absorbing set if for every deterministic bounded subset and a.e. , there exists such that for all ,
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 and every bounded set ,
Definition 2.2 A random compact set is called to be a random attractor for the RDS φ if is an attracting set and for and all .
Theorem 2.3 (see [4])
Assume thatis a continuous RDS onXover MDSθ. If there exists a compact random absorbing set, thenpossesses a random attractordefined by
wheredenotes all the bounded subsets ofX.
Let be the ptimes integrable functions space on D with norm denoted by , and with Sobolev equivalent norm (see p.166 of [14])
and . Let with the usual scalar product and norm . Then we have the following Gelfand triple
or concretely
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 selfadjoint, is the generator (with domain ) of a strongly continuous semigroup on which is contractive and positive. Here “contractive” means and “positive” means for every . The resolvent of generator Δ is denoted by , where is the resolvent set of Δ. By the LumerPhillips Theorem in [11], it follows that and for
Moreover, for and , where is the domain of Δ.
Since Δ is negative definite and selfadjoint, then Δ is associated with the Dirichlet forms by
is unique determined by Δ. For , we define a new inner product by
where is the resolvent of Δ. Then it follows from Ref. [8] that as , and
3 Existence and uniqueness of RDS
In this section, we show the existence and uniqueness of a continuous RDS for the following stochastic pLaplaciantype equation with multiplicative noise,
where , . To study System (3.1)(3.3), we assume that the nonlinearity defined in satisfies the following conditions:
For , we define a nonlinear operator A on V by
Then (3.1) reads
Since , by our assumption (3.4)(3.6) and , it is easy to check that for given , the operator mapping into is well defined, where .
Let be the probability space as in the introduction. Define the Wiener shift by
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 OrnsteinUhlenbeck process. Put
which solves the Itô differential equation
where the OrnsteinUhlenbeck constant equals to 1.
Note that is a Gaussian process with mathematical expectation and variance , see [7], whereas . Furthermore, from [2,15,18], the random variable is continuous in t for a.e. and grows sublinearly, i.e., .
We now translate (3.1) by one classical change of variables
Then we have
Then, formally, the variable satisfies the following equations parameterized by but without white noise:
where satisfies (3.4)(3.6) and f is given in , .
For convenience, we put
Then we have
Note that System (3.1)(3.3) and System (3.10)(3.12) are equivalent by (3.9). Let and 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 is continuous in H for the initial value in H, then the mapping 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, . Then for allwith, System (3.1)(3.3) has a unique solution
for allanda.e. . Furthermore, the mappingfromHintoHis continuous for all.
Proof We first show that for every there exists a unique solution . By Theorem 4.2.4 and Exercise 4.1.2 in [12], it suffices to show that for every fixed possesses Hemicontinuity, Monotonicity, Coercivity, and Boundedness 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 . By our assumptions that and , we can check that maps to for . Thus if , then (3.14) implies that . Now by the general fact (see p.164 of [14]) it follows that v is almost everywhere equal to a function belonging to . Hence by the transformation (3.9) and the continuous property of OrnsteinUhlenbeck process, is almost everywhere equal to a function belonging to .
We finally prove the continuity of the mapping from H into H. It suffices to prove that the mapping is continuous from H into H.
Let , be two different initial values at initial value time s, and corresponding solutions be denoted by and respectively. Then it follows from (3.14) that
where is defined in (3.13). Note that
Because the function is increasing for and , the last inequality in the above proof is correct. Then by a simple computation we find that for fixed ,
where is in (3.6). Hence, multiplying (3.15) by , integrating over D, and using (3.16), we get that
Using Gronwall’s lemma to (3.17) from s to t, it yields that
Then, the continuity of the mapping from H into H is followed from the contraction property (3.18). This finishes the total proofs of Theorem 3.1. □
We now define
with . By the uniqueness part of the solution in Theorem 3.1, we immediately get that is a stochastic flow; that is, for every and
Hence if we define
with , then by Theorem 3.1 ψ is a continuous RDS associated with System (3.1)(3.3).
We define
Then φ is a continuous RDS associated with System (3.10)(3.12), with the following fact
That is to say, can be interpreted as the position of the trajectory at time 0, which was in 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 and . Note that in the following ; the results will hold for a.e. and the generic constants c or , are independent of in the context, where .
Lemma 4.1Suppose thatgsatisfies (3.4)(3.6) andfis given in. Then there exist random radii, such that for allthere existssuch that for alland allwith, the following inequalities hold fora.e. ,
whereis the solution to Equation (3.10) with.
Proof For simplicity, we abbreviate for with . Multiplying both sides of (3.10) by and then integrating over D, we obtain that
where
Then by (4.1)(4.4), we have
Since , then by using Sobolev’s embedding inequality and inverse Young’s inequality we see that
Then it follows from (4.5) and (4.6) that
By employing Gronwall’s lemma over interval with , we find that
for . By the properties of the OrnsteinUhlenbeck process, we deduce that
and
Hence, given every fixed and , we can choose , depending only on ω and ϱ, such that for all and ,
which gives an expression for . Replacing t by τ in (4.5) and integrating for τ over intervals , then using (4.11) it yields that for all ,
Then we have
where
Thus the righthand side of (4.13) gives an expression for . □
In the following, we shall obtain the regularity of the solution to stochastic pLaplaciantype equation. This is the most challenging part in our discussion. Because of the nonlinearity of driven and function in Equation (3.10), it seems difficult to derive the Vnorm estimate as in [14], where the author only deals with a linear case, i.e., . So we relax to estimate the solution in a weaker Sobolev with equivalent norms denoted by for . 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. Then there exists a random radius, such that for allthere existssuch that for alland allwith, the following inequality holds fora.e.
whereis the solution to (3.10) with.
Proof Taking the inner product of (3.10) with where and , we get that
By the semigroup theory (see [14]) we have
for , the domain of Laplacian Δ. We now estimate all terms on the righthand side of (4.14). Employing (4.15) and integrating by parts, it yields that
where we use the contraction property of on , i.e.,
for and every . By our assumption (3.5), along with (4.17) for q, the second term on the righthand side of (4.14) is estimated as
where we employ Young’s inequality for twice. But, by Sobolev’s inequality and Young’s inequality, it yields that
and by (4.19) we have
Then by (4.18)(4.20), there exist positive constants c such that
For the third term on the righthand side of (4.14), by (4.17) we see that
On the other hand, by (4.15) and the Dirichlet forms (2.3), we have
Then it follows from (4.14), (4.16) and (4.21)(4.23) that
where
and c is a positive constant independent of λ. So taking limit on both sides of (4.24) for and associating with (2.2) and (2.4), we deduce that
Replacing t by τ in (4.25) and integrating τ from s to t (), it yields that
Put
Then by Lemma 4.1, (4.26) reads
Integrating (4.27) for s over intervals , we have
for all . By Poincare’s inequality and Young’s inequality, there exists positive constant c such that
Hence by using Lemma 4.1 again, along with (4.29), it follows from (4.28) that
with , which gives an expression for . 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. Then the RDSgenerated by System (3.10)(3.12) possesses a random attractordefined by
wheredenotes all the bounded subsets ofHand the closure is theHnorm.
5 The single point attractor
In this section, we consider a special case, that is, 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, . Then forand, there exists a positive constantsuch that
In particular, for each fixedandthere exists a single pointinHsuch that
for everybelonging to the bounded subsetBofH. Furthermore, the convergence in (5.1) is uniform with respect to all.
Proof Let be the solutions to (3.10) with initial values , . Then we can deduce from (3.14) that
Multiplying (5.2) by , integrating over D and using (3.16), we find that
Now, applying Gronwall’s lemma to (5.3) from to t, it yields that
We then estimate . By (4.5) we have
By the Sobolev’s embedding inequality and the inverse Young’s inequality, we can choose such that
So by (5.5) and (5.6) we get that
Using Gronwall’s lemma to (5.7) from to with , we get that
Similar to the argument of (4.10), we know that the integral in the last term on the righthand side of (5.8) is convergent. Hence, it follows from (5.8) and (5.4) that for every fixed ,
So for every bounded subset B of H and , it follows from (5.9) that
since by the properties of the OrnsteinUhlenbeck process, we have
and
and
Moreover, the convergence in (5.11)(5.13) is uniform with respect to , belonging to every bounded subset of H. Then (5.10) implies that for fixed , is a Cauchy sequence in H with respect to . Therefore, by the completeness of H, for every fixed and , has a limit in H denoted by , i.e.,
□
Theorem 5.2Assume thatgsatisfies (3.4)(3.6) andfis given in, . Then the RDSgenerated by the solution to (3.10)(3.12) possesses a single point attractor, i.e., there exists a single pointinHsuch that
Proof Put
Then is a stochastic flow associated with System (3.10)(3.12) and the RDS . By Lemma 5.1 we define
where . Then we need to prove that is a compact attractor. It is obvious that is a compact random set. Hence by Definition 2.2 it suffices to prove the invariance and attracting property for . Since by the continuity of and the flow properties of , we have
That is to say, . On the other hand, by the uniform convergence of (5.1), it follows from (3.22) that for every bounded subset ,
as . This shows that 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).
References

Arnold, L: Random Dynamical Systems, Springer, Berlin (1998)

Bates, PW, Lu, K, Wang, B: Random attractors for stochastic reactiondiffusion equations on unbounded domains. J Differential Equations. 246, 845–869 (2009). Publisher Full Text

Chen, G, Zhong, C: Uniform attractors for nonautonomous pLaplacian equations. Nonlinear Anal.. 68(11), 3349–3363 (2008). Publisher Full Text

Chueshov, I: Monotone Random Systems Theory and Applications, Springer, Berlin (2002)

Crauel, H, Debussche, A, Flandoli, F: Random attractors. J. Dyn. Differ. Equ.. 9(2), 307–341 (1997). Publisher Full Text

Crauel, H, Flandoli, F: Attractors for random dynamical systems. Probab. Theory Relat. Fields. 100(3), 365–393 (1994). Publisher Full Text

Fan, X: Attractors for a damped stochastic wave equation of SineGordon type with sublinear multiplicative noise. Stoch. Anal. Appl.. 24, 767–793 (2006). Publisher Full Text

Fukushima, M, Oshima, Y, Takeda, M: Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin (1994)

Li, Y, Guo, B: Random attractors for quasicontinuous random dynamical systems and applications to stochastic reactiondiffusion equations. J. Differ. Equ.. 245(7), 1775–1800 (2008). Publisher Full Text

Li, J, Li, Y, Wang, B: Random attractors of reactiondiffusion equations with multiplicative noise in . Appl. Math. Comput.. 215, 3399–3407 (2010). Publisher Full Text

Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983)

Prévôt, C, Röckner, M: A Concise Course on Stochastic Partial Differential Equations, Springer, Berlin (2007)

Schmalfuss, B: Backward cocycle and attractors of stochastic differential equations. In: Reitmann V, Riedrich T, Koksch N (eds.) International Seminar on Applied MathematicsNonlinear Dynamics: Attractor Approximation and Global Behavior, pp. 185–192. Technische Universität, Dresden (1992)

Temam, R: InfiniteDimensional Dynamical Systems in Mechanics and Physics, Springer, New York (1997)

Wang, B: Random attractors for the stochastic FitzHughNagumo system on unbounded domains. Nonlinear Anal.. 71, 2811–2828 (2009). Publisher Full Text

Wang, B: Asymptotic behavior of stochastic wave equations with critical exponents on . Trans. Am. Math. Soc.. 363, 3639–3663 (2011). Publisher Full Text

Wang, G, Guo, B, Li, Y: The asymptotic behavior of the stochastic GinzburgLandau equation with additive noise. Appl. Math. Comput.. 198(2), 849–857 (2008). Publisher Full Text

Wang, Z, Zhou, S: Random attractors for stochastic reactiondiffusion equations with multiplicative noise on unbounded domains. J. Math. Anal. Appl.. 384(1), 160–172 (2011). Publisher Full Text

Yang, M, Sun, C, Zhong, C: Existence of a global attractor for a pLaplacian equation in . Nonlinear Anal.. 66(1), 1–13 (2007). Publisher Full Text

Yang, M, Sun, C, Zhong, C: Global attractor for pLaplacian equation. J. Math. Anal. Appl.. 327(2), 1130–1142 (2007). Publisher Full Text

Zhang, Q: Random attractors for a GinzburgLandau equation with additive noise. Chaos Solitons Fractals. 39(1), 463–472 (2009). Publisher Full Text

Zhao, C, Duan, J: Random attractor for the Ladyzhenskaya model with additive noise. J. Math. Anal. Appl.. 362(1), 241–251 (2010). Publisher Full Text

Zhao, W, Li, Y: Existence of random attractor for a pLaplaciantype equation with additive noise. Abstr. Appl. Anal.. 2011, (2011)

Zhao, W, Li, Y: Asymptotic behavior of twodimensional stochastic magnetohydrodynamics equations with additive noises. J. Math. Phys.. 52, (2011)

Zhao, W, Li, Y: random attractors for stochastic reactiondiffusion equation on unbounded domains. Nonlinear Anal.. 75, 485–502 (2012). Publisher Full Text