Research Article | Open Access
Existence of Random Attractors for a -Laplacian-Type Equation with Additive Noise
We first establish the existence and uniqueness of a solution for a stochastic -Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.
Let , , be a bounded open set with regular boundary . In this paper, we investigate the existence of a solution and a random attractor for the following quasilinear differential equation influenced by additive white noise with the boundary conditions and the initial condition
In (1.1), , , are mutually independent two-sided real-valued Wiener processes, are given real-valued functions that will be assumed to satisfy some conditions. The unknown is a real-valued random process, sometimes denoted by or . The exterior forced function defined in is subjected to the following growth and monotonicity assumptions: where .
In deterministic case (without random perturbed term), if , Temam  proved the existence and uniqueness of the solution, and then obtained the global attractor. Recently, Yang et al. [2, 3] obtained the global attractors for a general -Laplacian-type equation on unbounded domain and bounded domain, respectively. Chen and Zhong  discussed the nonautonomous case where the uniform attractor was derived.
It is well known that the long-time behavior of random dynamical systems (RDS) is characterized by random attractors, which was first introduced by Crauel and Flandoli  as a generalization of the global attractors for deterministic dynamical system. The existence of random attractors for RDS has been extensively investigated by many authors, see [5–12] and references therein. However, most of these researches concentrate on the stochastic partial differential equations of semilinear type, such as reaction-diffusion equation [5–8], Ginzburg-Landau equation [9, 10], Navier-Stokes equation [5, 6], FitzHugh-Nagumo system  and so on. To our knowledge, recently, the Ladyzhenskaya model in  seems the first study on the random attractors for nonsemilinear type equations. It seems that the quasilinear type or complete nonlinear type evolution equations with additive noise take on severe difficulty when one wants to derive the random attractors.
In this paper, we consider the existence and uniqueness of the solution and random attractor for (1.1) with forced term satisfying (1.4)–(1.6). The additive white noise characterizes all kinds of stochastic influence in nature or man-made complex system which we must take into consideration in the concrete situation.
In order to deal with (1.1), we usually transform by employing a variable change the stochastic equation with a random term into a deterministic one containing a random parameter. Then the structure of the original equation (1.1) is changed by this transformation. As a result, some extra difficulties are developed in the process of the estimate of the solution, especially in the stronger norm space , where is the Gelfand triple; see Section 2. Hence, the methods (see [1–3]) used in unperturbed case are completely unavailable for obtaining the random attractors for (1.1).
Though we also follow the classic approach (based on the compact embedding) widely used in [5, 6, 9, 10, 12] and so on, some techniques have to be developed to overcome the difficulty of estimate of the solution to (1.1) in the Sobolev space . Fortunately, by introducing a new inner product over the resolvent of Laplacian, we surmount this obstacle and obtain the estimate of the solution in the Sobolev space , which is weaker than , see Lemma 4.2 in Section 4. Here some basic results about Dirichlet forms of Laplacian are used. For details on the Dirichlet forms of a negative definite and self-adjoint operator please refer to . The existence and uniqueness of solution, which ensure the existence of continuous RDS, are proved by employing the standard in . If a restrictive assumption is imposed on the monotonicity coefficient in (1.6) we obtain a compact attractor consisting of a single point which attracts every deterministic bounded subset of .
The organization of this paper is as follows. In the next section, we present some notions and results on the theory of RDS and Dirichlet forms which are necessary to our discussion. In Section 3, we prove the existence and uniqueness of the solution to the -Laplacian-type equation with additive noise and obtain the corresponding RDS. In Section 4, we give some estimates for the solution satisfying (1.1)–(1.6) in given Hilbert space and then prove the existence of a random attractor for this RDS. In the last section, we prove the existence of the single point attractor under the given condition.
In this section, we first recall some notions and results concerning the random attractor and the random flow, which can be found in [5, 6]. For more systematic theory of RDS we refer to . We then list the Sobolev spaces, Laplacian and its semigroup and Dirichlet forms.
The basic notion in RDS is a measurable dynamical system (MSD). The form is called a MSD if is a complete probability space and , is a family of measure-preserving transformations such that is measurable, and for all .
A continuous RDS on a complete separable metric space with Borel sigma-algebra over MSD is by definition a measurable map such that P-a.s. (i) on ,(ii) for all (cocycle property),(iii) is continuous for all .
A continuous stochastic flow is by definition a family of measurable mapping , such that P-a.s. for all , and is continuous in for all and .
A random compact set is a family of compact sets indexed by such that for every the mapping is measurable with respect to .
Let be a random set. One says that is attracting if for P-a.s. and every deterministic bounded subset where is defined by .
We say that absorbs if P-a.s. , there exists such that for all ,
Definition 2.1. Recall that a random compact set is called to be a random attractor for the RDS if for P-a.s. (i) is invariant, that is, , for all ;(ii) is attracting.
Theorem 2.2 (see ). Let be a continuous RDS over a MDS with a separable Banach Space . If there exists a compact random absorbing set absorbing every deterministic bounded subset of , then possesses a random attractor defined by where denotes all the bounded subsets of .
Let be the -times integrable functions space on with norm denoted by , be the space consisting of infinitely continuously differential real-valued-functions with a compact support in . We use to denote the norm closure of in Sobolev space , that is, . Since is a bounded smooth domain in , we can endow the Sobolev space with equivalent norm (see [1, page 166]) Define = the dual of , that is, . Then we have where . Let denote the closure of in with the usual scalar product and norm . Identifying with its dual space by the Riesz isomorphism , we have the following Gelfand triple: or concretely where the injections are continuous and each space is dense in the following one.
We define the linear operator by for . Then is negative definite and self-adjoint. It is well-known that (with domain ) generates a strongly continuous semigroup on which is contractive and positive. Here “contractive” means and “positive” means for every . The resolvent of generator denoted by , , where is the resolvent set of . By Lumer-Phillips Theorem in , it follows that and for Furthermore, by (2.10) for every we have where the convergence is in the -norms. Moreover, for , it follows that , and .
Since is negative definite and self-adjoint operator on , we associate with the Dirichlet forms  by is unique determined by . For , we define a new inner product by where is the resolvent of . Then, we have the basic fact (see ) that as , and for .
3. Existence and Uniqueness of RDS
We introduce an auxiliary Wiener process, which enables us to change (1.1) to a deterministic evolution equation depending on a random parameter. Here, we assume that is a two-sided Wiener process on a complete probability space , where , is the Borel sigma-algebra induced by the compact-open topology of and is the corresponding Wiener measure on . Then we have Define the time shift by Then is a ergodic measurable dynamical system.
In order to obtain the random attractor, in our following discussion, we always assume that belong to and .
We now employ the approach similar to  to translate (1.1) by one classical change of variables where, for short, . Then, formally, satisfies the following equation parameterized by : where satisfies (1.4)–(1.6) and is given in , .
We define a nonlinear operator on for , . Then we have where we define with as in (3.3). So we can deduce problem (1.1) to the problem with initial condition for . Moreover, by (3.9), it follows that the solutions -wise satisfy the following: with and .
Proof. We will show that possesses hemicontinuity, monotonicity, coercivity, and boundedness properties. Then for every with , the existence and uniqueness of solution follow from [13, Theorem 4.2.4]. If the solution , , then it is elementary to check that belongs to by our assumption and . Thus, from (3.9), we get that . Now by the general fact (see [1, page 164, line 1–3]) it follows that is almost everywhere equal to a function belonging to . The continuity of the mapping from into is easily proved by using the monotonicity of .
By [13, Theorem 4.2.4], it remains to show that possesses hemicontinuity, monotonicity, coercivity, and boundedness properties. For convenience of our discussion in the following, we decompose , where and , where is as in (3.7).
We first prove the hemicontinuity, that is, for every , the function is continuous from . But it suffices to prove the continuity at . So we assume that . For , by integration by parts, we see that By Hölder's inequality and Young's inequality, it yields that which the right-hand side is in for . Hence the expression of the right-hand side of inequality (3.13) is the control function for the integrant in (3.12). Then the Lebesgue's dominated convergence theorem can be apply to (3.12) when we take the limit . This proves the hemicontinuity of . As for the hemicontinuity of , noting that by our assumption (1.5) and we have It suffices to find the control function for the first integrand above, but we can get this by noting that and using approach similar to (3.13).
Second, we prove the monotonicity of . We first prove the monotonicity for . For , since , , we have Since , the function is increasing for , which shows that the last inequality in the above proof is correct. On the other hand, by our assumption (1.6), we have , and therefore it follows that for where is as in (1.6). Hence, we have showed the monotonicity of .
As for the coercivity, for , by our assumptions (1.4) and (1.5), using Hölder' inequality, we have where and are defined in (1.4) and (1.5). By employing the -Young's inequality, that is, for and , we find that Similarly, we have then, by (3.17)–(3.20), we obtain that with where is the dual number of . At the same time, (3.21) is one form of coercivity which will be used in Section 4, but in order to prove the existence and uniqueness of solution to (3.4), we will give another form.
Noting that by Hölder's inequality it follows with that then by the inverse -Young's inequality, that is, when and , we get from (3.23) that where and is the Sobolev embedding coefficient of . Hence, it follows from (3.21) that with , where is defined as in (3.22). Note that if , we omit this procedure and directly (3.21) passes to (3.25). Hence we have proved the coercivity for .
We finally prove the the boundedness for for fixed , that is, for fixed , is a linear bounded functional on . Indeed, for , by applying Hölder's inequality and repeatedly using Sobolev's embedding inequality, we have with the random variable and the positive constants independent of . Therefore, from (3.26) we finally find that is a bounded linear operator on for fixed . From the proof we know that the assumption is necessary. This completes the proof of Theorem 3.1.
We now define with . Then is the solution to (1.1) in certain meaning for every and . By the uniqueness part of 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 is a continuous stochastic dynamical system associated with quasilinear partial differential equation (1.1), 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 (see ).
4. Existence of Compact Random Attractor for RDS
In this section, we will compute some estimates in space and . Note that appearing in the proofs are given in Section 3. In the following computation, ; the results will hold for P-a.s. .
Lemma 4.1. Suppose that satisfies (1.4)–(1.6) and is given in . Then there exist random radii , such that for all there exists such that for all and all with , the following inequalities hold for P-a.s. where is the solution to (3.4) with and .
Proof. For simplicity, we abbreviate and for fixed and with . Multiplying both sides of (3.9) by and then integrating over , we obtain that
Then, by (3.25), we have
Applying the Gronwall's lemma to (4.3) from to , , it yields that
where grows at most polynomially as (see ). Since is multiplied by a function which decays exponentially, the integral in (4.4) is convergent.
Given every fixed , we can choose , depending only on and , such that . Similarly, grows at most polynomially as , and is multiplied by a function which decays exponentially. Then we have Hence by (4.4) we can give the final estimate for for . Following (4.2), by using (3.21), we find that where is the same as in (3.22). Integrating (4.7) for on , we get that which gives an expression for .
In the following, we give the estimate of . This is the most difficult part in our discussion. Because the nonlinearity of and in (3.4) or (3.9), it seems impossible to derive the -norm estimate by the way as [1, page 169]. So we relax to bound the solution in a weaker Sobolev with equivalent norms denoted by for . Here, just as our statement in the introduction, we make the inner product over the resolvent which is defined in Section 2, then by using the Dirichlet forms of we obtain technically the estimate of .
Lemma 4.2. Suppose that satisfies (1.4)–(1.6) and is given in . Then there exists a random radius , such that for all there exists such that for all and all with , the following inequality holds for P-a.s. where is the solution to (1.1) with . In particular,
Proof. Taking the inner product of (3.9) with where , , we get By the semigroup theory (see ) we have for . We now estimate every terms on the right-hand side of (4.11). The first term on the right-hand side of (4.11) is rewritten as Employing (4.12) and by integration by parts, it yields that where we use the contraction property of on , that is, for and every . We now bound the second term on the right-hand side of (4.13) where we use our assumption . Since maps into , then for fixed and every , we have So for fixed , , and therefore by (4.15) we obtain that where and is a constant independent of , and . Here we should note that is a bounded linear operator on . Hence, by (4.14)–(4.17) the first term on the right-hand side of (4.11) is finally bounded by By our assumption (1.5), the second term on the right-hand side of (4.11) is estimated as where we employ Young's inequality for . But, by Sobolev's inequality and Young's inequality, it yields that similarly where the positive constants are Sobolev's embedding constants independent of . Then by (4.19)–(4.21), there exist positive constants such that where . By (4.18) and (4.22), we find that (4.11) becomes where and . On the other hand, by (4.12) and the Dirichlet forms (2.14), we have Hence by (4.24), (4.23) is rewritten as Note that the right-hand side of (4.25) is independent of . So taking limit on both side of (4.25) for , association with (2.15), we deduce that Integrating (4.26) from to (), it yields that Therefore, by Lemma 4.1, we find that Integrating (4.28) for from −1 to 0, we have for all . By Poincare's inequality, and Young's inequality, there exist positive constants , , such that Hence, by (4.30) and using Lemma 4.1 again, (4.29) follows with . See that . Then, we have which gives an expression for . This completes the proof.
Theorem 4.3. Assume that satisfies (1.4)–(1.6) and is given in . Then the RDS generated by the stochastic equation (1.1) possesses a random attractor defined as where denotes all the bounded subsets of and the closure is the -norm.
Remark 4.4. As stated in Theorem 3.1, under the assumptions (1.4)–(1.6), the solutions of (1.1) are in . So it is possible in theory to obtain a compact random attractor in or . But it seems most difficulty to derive the estimate of solution in due to the nonlinear principle part .
5. The Single Point Attractor
In this section, we consider the attracting by a single point. In order to derive our anticipating result, we assume that in (1.6). This leads to the following fact that for every fixed and , the solution to (1.1) is a Cauchy sequence in for the initial time with initial value belonging to the bounded subset of . Then we obtain a compact attractor consisting of a single point which is the limit of as .
Lemma 5.1. Assume that satisfies (1.4)–(1.6) and is given in , . Then for and with and , there exists a positive constant such that In particular, for each fixed and there exists a single point in such that where and is the stochastic flow defined as in (3.28) which is a version of solution to (1.1). Furthermore, the limit in the above is independent of for all belonging to a bounded subset of .
Proof. For and with and , we can deduce from (3.9) and (3.8) that where is the solution to problem (1.1). On the other hand, by (3.8) and (3.16), we immediately deduce that Then, multiplying (5.3) by , integrating over , and using (5.4), we find that Now, applying Gronwall's lemma to (5.5) from to , it yields that We then estimate . To this end, we rewrite (3.21) as Since , by Hölder's inequality and inverse Young's inequality we can choose constant such that Then, by (5.7)-(5.8) it follows from (4.2) that where and . Using Gronwall's lemma to (5.9) from to with , we get that Similar to the argument of (4.4), we know that the integral in the above is convergent. Therefore, we have from which and (5.6) it follows for every fixed that where the convergence is uniform with respect to belonging to every bounded subset of . Then (5.12) implies that for fixed , is a Cauchy sequence in for . Thus, by the completeness of , for every fixed and , has a limit in denoted by , that is,
Theorem 5.2. Assume that satisfies (1.4)–(1.6) and is given in , . Then the RDS generated by the solution to (1.1) possesses a single point attractor , that is, there exists a single point in such that
Proof. By Lemma 5.1 we define where by (3.28). Then we need prove that is a compact attractor. It is obvious that is a compact random set. Hence by Definition 2.1 it suffices to prove the invariance and attracting property for . Since by the continuity of , and relations (3.29)–(3.32), we have