RANDOM ATTRACTOR OF NONLINEAR STRAIN WAVES WITH WHITE NOISE

In this paper, we consider the long time behaviors of nonlinear strain waves in elastic waveguides with white noise. We show that the initial boundary value problem has a global solution and a compact global attractor.


Introduction
In some problems of nonlinear wave propagation in waveguides, the interaction of waveguides, the external medium and the possibility of energy exchange through lateral surface of waveguide cannot be neglected. When the energy exchange between the rod and the medium is considered, there is a dissipation of deformation wave in the viscous external medium. The general cubic double dispersion equation (CDDE) can be derived from Hamilton principle: where a, b, c, d are some positive constants depending on Young modulus E 0 . The equation (1.1) was studied in the literatures [5,6,12,13,14,15]. In this paper, we consider the following stochastic nonlinear wave equation perturbed by a random forcing term du t − (αdu t + γdu) xx = (u − βu xx + f (u)) xx dt + g(x)dt + m j=1 h j dw j , (1.2) where α, β, γ are positive constants, f is a sufficiently smooth real valued function with f (0) = 0, g and h j (j ∈ {1, 2, · · ·, m}) are given functions defined on R and {w j } m j=1 are independent two side real-valued Wiener processes on a probability space which will be specified later.
Attractor is an important concept in the study of asymptotic behavior of deterministic dynamical system. Crauel, Debussche and Flandoli [4] present a general theroy to study the random attractor by defining an attracting set as a set that attracts any orbit starting from −∞. The random attractors are compact invariant
A set-valued map B : Ω → 2 X is called a random closed set if B(ω) is a nonempty closed set and ω → d(x, B(ω)) is measurable for x ∈ X. A random set B(ω) is called tempered if for P-a.s. ω ∈ Ω and all β > 0 Let D be the collection of all tempered random subsets in X and {K(ω)} ω∈Ω ∈ D. Then {K(ω)} ω∈Ω is called a random absorbing set for S in D if for B(ω) ∈ D and P -a.e. ω ∈ Ω, there exists t B (ω) > 0 such that Definition 2.2. A random set {A(ω)} ∈ D is random attractor (or pullback attractor) for a RDS S if the following conditions are satisfied, for P -a.e. ω ∈ Ω, (i) A(ω) is a random compact set. i.e. ω → d(x, A(ω)) is measurable for every x ∈ X and A(ω) is compact; S(t, ω, A(ω)) = A(ϑ t ω) for all t ≥ 0; (iii) {A(ω)} attracts every set in D, i.e., for all B = {B(ω)} ∈ D, where d H is the Hausdorff semi-distance.
Let B be a bounded set in a Banach space X. The Kuratowski measure of non-compactness α(B) of B is defined by α(B) = inf{d > 0 : B admits a f inite cover by sets of diameter ≤ d}.

The basic setting and O-U processes
In this section, we present the existence of continuous random dynamical system for the stochastic nonlinear strain wave equation in elastic waveguides: subject to the initial conditions and boundary condition u(0, t) = u(l, t) = 0, (3.3) where ∆ = ∂ xx , α, β, γ are positive constants, g is a given function in L 2 (0.l), for j ∈ {1, 2, · · ·, m}, h j ∈ H 1 0 ∩ W 2,q (0, l) for some q ≥ 2 and {w j } m j=1 are independent two-sided real valued Wiener processes on a probability space, which will be specified below and f is a nonlinear function satisfying the following conditions: for all s ∈ R where F (s) = s 0 f (τ )dτ and c i (i = 1, 2, 3) are positive constants. In the sequel, we consider the probability space (Ω, F , P ), where the Borel σ−algebra F on Ω is generated by the compact open topology, and P is the corresponding Wiener measure on F . Then we identify ω(t) with (w 1 , w 2 , · · ·, w m ), i.e., The time shift is defined by It is a family of ergodic terms formations. Now we consider the one-dimensional Ornstein-Uhlenbeck equation It is easy to check that for each j = 1, 2, · · ·, m is a solution of (3.6). Putting z( where p ≥ 0 and ρ 1 (ω), ω ∈ Ω satisfies Proof. Let j = 1, 2, · · ·, m. Since |z j (ω j )| is a tempered random variable and the mapping t → ln |z j (ϑ t ω j )| is P-a.s.continuous, it follows from Proposition 4.3.3 in [13] that for any ǫ j > 0, there is a tempered random variable r j (ω j ) > 0 such that |z j (ω j )| ≤ r j (ω j ) and Let ǫ > 0 and ǫ 1 = ǫ 2 = · · · = ǫ m = ǫ, then we have Let ρ 1 (ω) = m j=1 r j (ω j )||h j || 2p+2 then (3.8) holds and (3.9) follows from (3.10).
for t ∈ R and ω ∈ Ω, where λ 1 is the first eigenvalue of −∆.
By the Poincare inequality the corollary holds. Now transform the problem (3.1)-(3.3) to a deterministic system with a random parameter and show that it generates a random dynamical system.
Let v(t, ω) = u t (t, ω) + εu(t, ω) − z(ϑ t ω). Then (3.1)-(3.3) is equivalent to the following random partial differential system where . Then E 1 ֒→ E 0 with compact imbedding. By a Galerkin method as in [6], it can be proved that under assumptions (3.4) and (3.5), for P-a.e.ω ∈ Ω and for every (u 0 , v 0 ) ∈ E 0 , problem (3.12)-(3.15) have a unique solution (u, v) ∈ C(R + , E 0 ) and the solution (u, v) is continuous with respect to x in E 0 for all t ≥ 0. Hence, the solution mapping generates a RDS. It is called stochastic flow associated with the nonlinear strain wave equation with additive noise.

Uniform time a priori estimates and random attractors
In this section, we derive uniform estimates on the solutions of (3.12)-(3.15) when t → ∞ and prove the existence of a bounded random absorbing set and the asymptotic compactness of the random dynamical system associated with the equation. From now on, we always assume that D is the collection of all tempered subsets of E 0 with respect to (Ω, F , P, (ϑ t ) t∈R ). Let E 0 = H 1 0 ×L 2 (0, l) endowed with the inner product and norm (Y 1 , . We first derive the following uniform estimates in E 0 . ) is a positive random function. Proof. Taking the inner product of (3.14) with (−∆) −1 v and using
Proof. Taking the inner product of (4.18) with (−∆) −1 v a and using v a = u a t +εu a , we have d dt and where λ = δε > 0. Applying Gronwall's lemma, we obtain for all t ≥ 0

By (4.4), we have
By arguments similar to (4.6), we can derive that where a, b are same as in Lemma 4.1. Replacing ω by ϑ −t ω with t ≥ 0 in (4.19), implies that the result holds.
We are now in a position to present our main result: Theorem 4.4. Assume that g ∈ H −1 (0, l) and (3.4) and (3.5) hold. Then the random dynamical system S(t, ω) has a unique random attractor in E 0 .
Proof. By Lemma 2.4, Lemma 4.2 and Lemma 4.3, the stochastic dynamical system S(t, ω) of nonlinear strain waves is almost surely D − α−contracting. This together with Lemma 2.5 implies that the existence of a unique D−random attractor for S(t, ω).