Asymptotic Behavior of Stochastic Wave Equations with Critical Exponents on R^3

The existence of a random attractor in H^1(R^3) \times L^2(R^3) is proved for the damped semilinear stochastic wave equation defined on the entire space R^3. The nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. The uniform pullback estimates on the tails of solutions for large space variables are established. The pullback asymptotic compactness of the random dynamical system is proved by using these tail estimates and the energy equation method.


Introduction
This paper deals with the existence of a random attractor for the stochastic wave equation defined on R 3 : with the initial conditions u(x, τ ) = u 0 (x), u t (x, τ ) = u 1 (x), (1.2) where x ∈ R 3 , t > τ with τ ∈ R, α and λ are positive numbers, g and h are given in L 2 (R 3 ) and H 1 (R 3 ) respectively, f is a nonlinear function with cubic growth rate (called the critical exponent), and w is an independent two-sided real-valued Wiener process on a probability space.
The global attractors of deterministic wave equations have been studied extensively in the literature, see, e.g., [3,4,18,24,27] and the references therein. Particularly, the existence of attractors was proved in [2,3,4,12,16,19,25,26] for the deterministic wave equations defined on bounded domains with critical exponents, and in [13,14,15,23] for the equations defined on unbounded domains with critical or supercritical exponents. In this paper, we will prove the existence of a random attractor for the stochastic wave equation ( To study the long term behavior of solutions of stochastic differential equations, the concept of random attractor should be used instead of global attractor, which was introduced in [11,17] for random dynamical systems. Since the nonlinearity f of equation (1.1) has a critical growth rate, the mapping f from H 1 (Q) to L 2 (Q) is continuous, but not compact, even for a bounded domain Q in R 3 . To circumvent the difficulty and prove the asymptotic compactness of the deterministic wave equation on a bounded domain Q, an energy equation approach was developed by Ball in [4]. This method is quite effective for a variety of applications, see, e.g., [5,20,21,22,30]. Notice that the compactness of embeddings H 1 (Q) ֒→ L p (Q) with p < 6 was crucial and frequently used in [4] when Q is bounded. In our case, the domain R 3 is unbounded, and hence the embeddings are not compact for any p. This means that Ball's method [4] alone is not sufficient for proving the asymptotic compactness of the equation on R 3 . We must overcome the difficulty caused by the non-compactness of embeddings H 1 (R 3 ) ֒→ L p (R 3 ) for p < 6. In this paper, we will solve the problem by using the method of tail estimates developed in [28] for deterministic parabolic equations. In other words, we will first show that the solutions of problem (1.1)- (1.2) uniformly approach zero, in a sense, as x and t go to infinity, and then apply these tail estimates and the energy equation method [4] to prove the asymptotic compactness of the stochastic wave equations on R 3 .
The random attractors of stochastic equations have been investigated by several authors in [1,8,9,10,11,17] and the references therein. In these papers, the domains of PDEs were supposed to be bounded. In the case of unbounded domains, the existence of random attractors has been established recently for parabolic and wave equations in [7] and [29], respectively. Notice that the method of [29] only works for the wave equation with subcritical nonlinearity, and is not valid for the critical case. It is the intension of this paper to prove the existence of a random attractor for the stochastic wave equation with critical nonlinearity on R 3 .
This paper is organized as follows. In the next section, we recall the random attractors theory for random dynamical systems. In Section 3, we define a continuous random dynamical system for problem (1.1)-(1.2). The uniform estimates of solutions are contained in Section 4, which include uniform estimates on the tails of solutions. In Section 5, we prove the pullback asymptotic compactness and the existence of random attractors for the stochastic wave equation on R 3 .
In the sequel, we adopt the following notations. We denote by · and (·, ·) the norm and the inner product of L 2 (R 3 ), respectively. The norm of a given Banach space X is written as · X . We also use · p to denote the norm of L p (R 3 ). The letters c and c i (i = 1, 2, . . .) are generic positive constants which may change their values from line to line or even in the same line.

Preliminaries
In this section, we recall some basic concepts related to random attractors for stochastic dynamical systems. The reader is referred to [1,6,10,17] for more details.
Let (X, · X ) be a separable Hilbert space with Borel σ-algebra B(X), and (Ω, F, P ) be a probability space.
Definition 2.2. A continuous random dynamical system (RDS) on X over a metric dynamical system (Ω, F, P, (θ t ) t∈R ) is a mapping which is (B(R + ) × F × B(X), B(X))-measurable and satisfies, for P -a.e. ω ∈ Ω, is the identity on X; Hereafter, we always assume that Φ is a continuous RDS on X over (Ω, F, P, (θ t ) t∈R ).
Definition 2.4. A random function r(ω) is called tempered with respect to (θ t ) t∈R if for P -a.e.
Definition 2.5. Let D be a collection of random subsets of X. Then D is called inclusion-closed Definition 2.6. Let D be a collection of random subsets of X and {K(ω)} ω∈Ω ∈ D. Then {K(ω)} ω∈Ω is called an absorbing set of Φ in D if for every B ∈ D and P -a.e. ω ∈ Ω, there exists Definition 2.7. Let D be a collection of random subsets of X. Then Φ is said to be D-pullback asymptotically compact in X if for P -a.e. ω ∈ Ω, {Φ(t n , θ −tn ω, x n )} ∞ n=1 has a convergent subsequence in X whenever t n → ∞, and x n ∈ B(θ −tn ω) with {B(ω)} ω∈Ω ∈ D.
Definition 2.8. Let D be a collection of random subsets of X and {A(ω)} ω∈Ω ∈ D. Then {A(ω)} ω∈Ω is called a D-random attractor (or D-pullback attractor) for Φ if the following conditions are satisfied, for P -a.e. ω ∈ Ω, where d is the Hausdorff semi-metric given by d(Y, Z) = sup y∈Y inf z∈Z y − z X for any Y ⊆ X and Z ⊆ X.
The following existence result on a random attractor for a continuous RDS can be found in [6,17]. Proposition 2.9. Let D be an inclusion-closed collection of random subsets of X and Φ a continuous RDS on X over (Ω, F, P, (θ t ) t∈R ). Suppose that {K(ω)} ω∈K is a closed absorbing set of Φ in D and Φ is D-pullback asymptotically compact in X. Then Φ has a unique D-random attractor {A(ω)} ω∈Ω which is given by In this paper, we will denote by D the collection of all tempered random sets of H 1 (R 3 )× L 2 (R 3 ), and prove problem (1.1)-(1.2) has a D-random attractor.

Random Dynamical Systems
In this section, we define a continuous random dynamical system for problem (1.1)-(1.2). Denote by z = u t + δu where δ is a small positive number to be determined later. Substituting u t = z − δu into (1.1) we find that du dt with the initial conditions where z 0 (x) = u 1 (x) + δu 0 (x), x ∈ R 3 , t > τ with τ ∈ R, α and λ are positive numbers, g ∈ L 2 (R 3 ) and h ∈ H 1 (R 3 ) are given, and w is an independent two-sided real-valued Wiener process on a complete probability space (Ω, F, P ) with path ω(·) in C(R, R) satisfying ω(0) = 0. In addition, (Ω, F, P, (θ t ) t∈R ) forms a metric dynamical system, where (θ t ) t∈R is a family of measure preserving shift operators given by θ t ω(·) = ω(· + t) − ω(t), ∀ ω ∈ Ω and t ∈ R.
Let F (x, u) = u 0 f (x, s)ds for x ∈ R 3 and u ∈ R. We assume the following conditions on the the nonlinearity f , for every x ∈ R 3 and u ∈ R, where 1 ≤ γ ≤ 3. As a special case, γ = 3 is referred to as the critical exponent. Notice that (3.4) and (3.5) imply which is useful when deriving uniform estimates of solutions.
To study the dynamical behavior of problem (3.1)-(3.3), we need to convert the stochastic system into a deterministic one with a random parameter. To this end, we set v(t, τ, ω) = z(t, τ, ω)− hω(t).
Then it follows from (3.1)-(3.3) that with the initial conditions By a standard method as in [13], it can be proved that problem (3.9)-(3.11) with (3.4)-(3.7) is well- . Sometimes, we also write the solution as (u(t, τ, ω, u 0 ), v(t, τ, ω, v 0 )) to indicate the dependence of (u, v) on initial data (u 0 , v 0 ). The following weak continuity of solutions on initial data is useful when proving the asymptotic compactness of solutions in the last section.
We now define a random dynamical system for the stochastic wave equation. Let Φ be a mapping, . Then Φ is a continuous . It is easy to verify that Φ satisfies the following identity, for P -a.e. ω ∈ Ω and t ≥ 0, Throughout this paper, we always denote by D the collection of all tempered random subsets of , and will prove Φ has a D-random attractor.

Uniform Estimates
In this section, we derive uniform estimates on solutions of problem (3.9)-(3.11). These estimates are needed for proving the existence of random absorbing sets and the pullback asymptotic compactness of the random dynamical system Φ.
Let δ > 0 be small enough such that and denote by where c 2 is the positive constant in (3.5).
where R(ω) is a positive tempered random function.
Proof. Taking the inner product of (3.10) with v in L 2 (R 3 ), we get and It follows from (4.5)-(4.8) that We now estimate every term on the right-hand side of (4.9). For the first term, by (4.1) we have The second term on the right-hand side of (4.9) satisfies For the third term on the right-hand side of (4.9), by (3.4) and (3.6), we obtain Similarly, by Young's inequality, the last two terms on the right-hand side of (4.9) are bounded by (4.14) By (4.9)-(4.14), we find that By (3.6) and (4.2) we have which along with (4.15) implies that Notice that r(ω) is well defined since ω(ξ) has at most linear growth rate as |ξ| → ∞. By and (4.17) and (4.19) we obtain that, for all τ ≤ T and t ∈ [τ, 0], By (3.6), we find that, for all t ≤ 0, By (4.20) and (4.21) we have that, for all τ ≤ T and t ∈ [τ, 0], which implies (4.3) and (4.4) with R(ω) = c(1 + r(ω)). Next we show that R(ω) is tempered, that is, for every β > 0, we want to prove Without loss of generality, we now assume β ≤ σ. Then we have Then (4.22) follows from (4.23) since ω has at most linear growth rate at infinity. This completes the proof.
We now derive an energy equation for problem (3.9)-(3.11). To this end, denote by, for (u, v) ∈ and Ψ(u(t, τ, ω, u 0 ), v(t, τ, ω, v 0 )) The energy equation (4) will be used to prove the pullback asymptotic compactness of solutions in the last section.
In what follows, we derive uniform estimates on the tails of solutions when x and t approach infinity. These estimates will be used to overcome the difficulty caused by non-compactness of embeddings H 1 (R 3 ) ֒→ L p (R 3 ) for p ≤ 6, and are crucial for proving the pullback asymptotic compactness of the random dynamical system. Given k ≥ 1, denote by Q k ={x ∈ R 3 : |x| < k} and Then for every ǫ > 0 and P -a.e. ω ∈ Ω, there exist T = T (B, ω, ǫ) < 0 and k 0 = k 0 (ω, ǫ) > 0 such that for all τ ≤ T and k ≥ k 0 , the solution (u(·, τ, ω, u 0 ), v(·, τ, ω, v 0 )) of problem (3.9)-(3.11) with Proof. Take a smooth function ρ such that 0 ≤ ρ ≤ 1 for all s ∈ R and Then there is a positive constant c such that |ρ ′ (s)| ≤ c for all s ∈ R.

Random Attractors
In this section, we prove existence of a D-random attractor for the stochastic wave equation on R 3 .
We first show that the random dynamical system Φ has a closed random absorbing set in D, and then prove that Φ is D-pullback asymptotically compact.
This completes the proof.
As an immediate consequence of Lemma 5.1, we see that the random dynamical system Φ is pullback asymptotically compact in H 1 (R 3 ) × L 2 (R 3 ).
We are now in a position to prove existence of a random attractor for the stochastic wave equation.