Asymptotic Behavior of Stochastic Strongly Wave Equation on Unbounded Domains

We study the asymptotic behavior of solutions to the stochastic strongly damped wave equation with additive noise defined on unbounded domains. We first prove the uniform estimates of solutions, and then establish the existence of a random attractor.


Let ( )
, , Ω   be a probability space, where , , , , : 0 0 the Borel σ -algebra  on Ω is generated by the compact open topology (see [1]), and  is the corres- ponding Wiener measure on  .Define ( ) , , , t t θ ∈ Ω    is an ergodic metric dynamical system.Consider the following stochastic strongly damped wave equation with additive noise defined in the entire space n  ( ) with the initial value conditions ,0 , ,0 , where ∆ is the Laplacian with respect to the variable Ω   .We identify ( ) , , , m W t W t W t  , i.e., ( ) ( ) ( ) ( ) ( ) Many works have been done regarding the dynamics of a variety of systems related to Equation (1).For example, the asymptotical behavior of solutions for deterministic strongly damped wave equation has been studied by many authors (see [2]- [11], etc.).For stochastic wave equation, the asymptotical behavior of solutions have been studied by several authors (see [12]- [25], etc.).However, no results have been presented on random attractors for stochastic strongly damped wave equation (1) with additive noise on unbounded domains to date.
In general, the existence of global random attractor depends on some kind compactness (see, e.g., [26]- [30]).For Cauchy problem, the main question is how to overcome the difficulty of lacking the compactness of Sobolev embedding in unbounded domains.For some deterministic equations, the difficulty caused by the unboundedness of domains can be overcome by the energy equation approach.The energy equation method was developed by Ball in [31] [32] and used by many authors (see, e.g., [33]- [39]).Under certain circumstances, the tail-estimates method can be used to deal with the problem caused by the unboundedness of domains (see [40]).In this paper, we will combine the splitting technique in [20] with the idea of uniform estimates on the tails of solutions to investigate the existence of global attractor of the stochastic strongly damped wave Equation (1) defined on unbounded domains.The rest of this paper is organized as follows.In the next section, we recall some basic concepts related to random attractor for general random dynamical systems.In Section 3, we provide some basic settings about Equation (1) and show that it generates a random dynamical system, and then we prove the uniform estimates of solutions and obtain the existence of a random attractor for Equation (1).
Throughout this paper, we use ⋅ and ( ) , ⋅ ⋅ to denote the norm and the inner product of ( ) L  , respectively.The norm of a Banach space X is generally written as X ⋅ .The symbol c is a positive constant which may change its value from line to line.

Preliminaries
In this section, we collect some basic knowledge about general random dynamical systems (see [1] [41] for details).Let ( ) , X X ⋅ be a separable Hilbert space with Borel σ -algebra ( ) , , , t t θ ∈ Ω    be the metric dynamical system on the probability space ( ) , , Ω   .
In the following, a property holds for  -a.e.ω ∈ Ω means that there is 0 Ω ∈ Ω with ( )

{ }
D ω is said to be bounded if there exist 0 u X ∈ and a random variable ( ) 0 lim e 0 for all 0

{ }
B ω is said to be a random absorbing set if for any tempered random set ( )

{ }
D ω , and  -a.e.ω ∈ Ω , there exists ( ) B ω is said to be a random attracting set if for any tempered random set ( )

{ }
D ω , and  -a.e.ω ∈ Ω , we have where H d is the Hausdorff semi-distance given by ( ) • ϕ is said to be asymptotically compact in X if for  -a.e.ω ∈ Ω , ( ) has a convergent subsequence in X whenever n t → +∞ , and ( )

{ }
A ω is said to be a random attractor if it is a random attracting set and for  -a.e.ω ∈ Ω and all 0 t .Theorem 1 (See [41]) Let ϕ be a continuous random dynamical system with state space X over ( ) ( )

{ }
A ω is a random attractor of ϕ , where

{ }
A ω is the unique random attractor of ϕ .

Basic Settings
In this subsection, we outline some basic settings about (1)-( 2) and show that it generates a random dynamical system.Let where σ is a small positive constant whose value will be determined later, then (1)- (2)   can be rewritten as the equivalent system with the initial value conditions where ( ) ( ) ( ) x ∈  and u ∈  .The function f will be assumed to satisfy the following conditions, (F1) , i c i = are positive constant.Note that (F1) and (F2) imply ( ) ( ) For our purpose, it is convenient to convert the problem (3)-(4) (or (1)-( 2)) into a deterministic system with a random parameter, and then show that it generates a random dynamical system.Let ( ) ( ) where ( ) Then it follows from the above, for  -a.e.ω ∈ Ω , ( ) ( ) Put ( ) ( ) , we obtain the equivalent system of (3)-( with the initial value conditions where ( ) ( ) ( ) x ∈  .We will consider (9)- (10) for ω ∈ Ω  and write Ω  as Ω from now on.Let ( ) ( ) where ⋅ denotes the usual norm in ( ) L  and Τ stands for the transposition.
By a standard method as in [2] [3] [42], one may show that under conditions (F1)-(F4), for ( ) Τ which is continuous with respect to ( ) in E for all 0 t > .Hence, the solution mapping generates a continuous random dynamical system, where ( ) . Introducing the homeomorphism also generates a random dynamical system associated with (3)-( 4).Note that the two random dynamical systems are equivalent.By (13), it is easy to check that ( ) Then, we only need to consider the random dynamical system ( )

Uniform Estimates of Solutions
In this subsection, we derive uniform estimates on the solutions of the stochastic strongly damped wave Equations (3)-( 4) defined on n  when t → ∞ .These estimates are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the random dynamical system associated with the equations.In particular, we will show that the tails of the solutions for large space variables are uniformly small when time is sufficiently large.
We assume that D is the collection of all tempered random subsets of E from now on.Let ( ) where 2 c is the positive constant in (F2).We define a new norm for 11).The next lemma shows that ( ) Then there exists a random ball Proof.Taking the inner product of the second equation of ( 9) with v in ( ) .
By the first equation of ( 9), we have ( ) Then substituting the above v into the second and third terms on the left-hand side of ( 17), we find that Using the Cauchy-Schwartz inequality and the Young inequality, we have , 4 By ( 19)- (24), it follows from (17) that ( ) .
Recalling the new norm E ⋅ in (15), by ( 14) we obtain from ( 25) that Using the Gronwall lemma, we have ( ) Substituting ω by t θ ω − , then we have from ( By ( 5), we get By assumption, ( )

{ }
B ω ∈D is tempered.Then, by (29), if ( ) ( ) Note that ( ) ( ) By (F3), we have that Combining ( 28), ( 30), ( 31) and (32), there is a where So, the proof is completed.To prove asymptotic compactness of the random dynamical system ( ) , S t σ ω , we first prove that the solu- tions were uniformly small outside a bounded domain and then decomposed the solutions in a bounded domain in terms of eigenfunctions of negative Laplacian as in [20].
Given 1 r , denote by { } Choose a smooth function ρ , such that ( ) ( ) and there exist constants 5 6 , c c , such that  , such that the so- lution ϕ of (9)-( 10) satisfies for  -a.e.ω ∈ Ω , t T Proof.We first consider the random Equations ( 9)- (10).Then taking the inner product of the second equation of ( 9) with Substituting v in (18) into the third, fourth and fifth terms on the left-hand side of (36), we get that By using conditions (F1), (F2) and (F3), we find By the Cauchy-Schwartz inequality and the Young inequality, we obtain ( ) ( ) Then it follows from ( 37)-( 42) that ( then, by (14) we have from (43) that .
By using the Gronwall lemma, we get that By using (F3), there exists ( ) In what follows, we estimate the terms on the right-hand side of (47).By ( 5), ( ) ( )

{ }
B ω is tempered, we have that, there exists ( ) , such that for all Since ( ) the second term on the right-hand side of (47) satisfies Note that ( ) , there is ( ) , such that for all 3 r R   , the third term on the right-hand side of ( Next, we estimate the forth term on the right-hand side of (47).Using (F3), replacing t by s and then ω by ) )     , then, combining (48), ( 49), ( 50), ( 51) and (54), we have for all t T >  and r R >  , ( ) ( ) Then we complete the proof.
Let ˆ1 ρ ρ = − with ρ given by ( 35) and denote by Multiplying ( 9) by ˆr ρ and using (57) we find that ( ) The problem has a family of eigenfunctions { } i i e ∈ with the eigenvalues { } , : be the projection operator.Lemma 3 Assume that (F1)-(F4), ( )  L  , we have ( , . Substituting ,2 ˆn v in (61) into the the third, fourth and fifth terms on the left-hand side of (62), we have , d Using conditions (F1) and (F4), we have ˆ, , By using the Cauchy-Schwartz inequality and the Young inequality, we have We next estimate each term on the right-hand side of (77).Since given; f is a nonlinear function satisfying certain dissipative and growth conditions, and { } 1 t