Random Attractors For A Wave Equations With Sublinear Multiplicative Noise

In this paper, we discuss the existence of a compact random attractor for such RDS which generated by a wave equation with sublinear multiplicative noise. Attractor of RDS is one of the interesting problems in the world recently.Early in the year of1994, Hans Crauel and Franco Flandoli described the notion of RDS in [4]:contained RDS、attracting set、absorbing set and global attractor, where the global attractor is a compact random invariant set which attracts all deterministic bounded sets; and pointed out that if there exists a random compact set absorbing every deterministic bounded sets, then this RDS possesses a global attractor. Thus to proof a RDS possesses a global attractor is only to proof this RDS existence of a random compact set absorbing every deterministic bounded sets.In this paper, consider following wave equation with sublinear multiplicative white noise: Where Ω is an open bounded set of Rn,(n=1,2,3) with a sufficiently regular boundary (?)Ω.The unknown function u=u(x,t) is a real-valued function on Ω×[τ,+∞),τ∈R. α,λ are positive, u0(x)∈H01(Ω), u1(x)∈L2(Ω). f(x) is given and independent of time, and f(x)∈H2(Ω)∩H01(Ω)。W(t) is a two-sided Wiener process, while c(u) o dW(t) describes sublinear multiplicative white noise.Through the proof in this paper, the RDS that is generated by the former equation is really existing a global attractor. This paper is divided into four chapters:Chapter one: the background on the theory of RDS and global attractor. And some preliminary definitions and results that will be used in this dissertation are presented.Chapter two: described the form of equation in details. Redefine some spaces, and pointed out H0γ1(Ω) and H01(Ω) are equivalent、D(A1+σ/2)γ and D(A1+σ/2) are equivalent, so E=H0γ1(Ω)×L2(Ω) and E’=01(Ω)×L2(Ω) are equivalent, Eσ=D(A1+σ/2)γ×D(Aσ/2) and Eσ‘=D(A1+σ/2)×D(Aσ/2) are equivalent. Put out theorem2.2.1, show the equation (2.2.1) is really existing a unique solution in the spaces of E and Eσ respectively when the initial conditions u(x,τ)=u0(x), ut(x,τ)=u1(x) satisfy each situation. Finally pointed out the solution of the equation (2.2.1) generated a RDS (2.3.1).Chapter three: first proof the existence of absorbing set in E.Proposition3.1.2B is any bounded set, when the μ in the Definition1.3.8is large enough, then there exist a random variable p(ω)>0, for any (u(x,τ),v(x,τ))T∈B, TB(ω)≥0, such that when-τ>TB(ω), the following hold P-a.s. for any ω∈Ω‖φ(0,θτω;φτ(ω))‖E≤ρ(ω)Second proof the existence of absorbing set in Eσ.Proposition3.2.2B is a bounded subset of E, for large enough μ, there exists a random variable ρ1(ω)>0, such that φi,φj,φk satisfy respectively: Where σ is as (3.2.2). Let D(ω) is an open ball with ρ1(ω) as its random radius in the E, so when σ> max{1,α,64k12?(1+1/ε)2,16k22(1+1/ε)2,64c1/ε2}, for any (u0,u1+εu0)T∈B, and large enough-τ, the following holdFinally shows the existence of the compact random attractor in E.Because Eσ is a compact subspace of E, D(ω) is a compact subset of E. because of Theorem1.3.1, Proposition3.1.2and Proposition3.2.2, we have: Theorem3.2.1when σ> max{1,α,64k12·(1+1/ε)2,16k22(1+1/ε)2,64c1/ε2}, RDS (2.3.1) has a compact random attractor.Chapter four:More study about this equations are needed.