Random Attractors Of Semilinear Parabolic Equation With Multiplicative Noise

In this paper,We will consider the following generalized Ginzburg-Landau equation with Multiplicative Noise:du+kuxdt=(u+γuxx-β|u|2u-δ|u|4u-μ|u|2ux-vu2ux)dt+uodW(t).where α=α1+iα2,β=β1+iβ2;γ=γ1+iγ2,μ=μ1+iμ2,v=v1+iv2are complex. α1>0,γ1>0.We assume that γ1>0,δ1>0且δ1γ1>|μ|2+|v|2.The white noise described by a process W(t),which is a Wiener process on the prob-ability space (Ω,F,P),where Ω={ω∈C(R,R):ω(0)=0}.F is the Borel Sigmaalgebra induced by the compact-open topology of Ω,P is a Wiener measure.This is supplemented with the boundary and initial conditions: u(0,t)=u(1,t)=0,t≥0u(x,t0)=u0(x),x∈R1.This paper is arranged as follows,we solve the stochastic generalized Ginzburg-Landau equation and get the corresponding RDS. Then, the existence of random attractors for this RDS in H is established.We mainly prove the existence and uniqueness of solution of the equation with the boundary condition and the initial condition,and the existence of the random attractors of the random dynamical system(RDS) generated by the unique solution.This paper is divided into four chapters: In chapter one:the background on the theory of RDS, global attractor and gener-alized Ginzburg-Landau equation,and some results that will be used in this dissertation are presented.In Chapter two: There is a unique solution of the equations(2.1.1)-(2.1.3) u=u(t,ω;t0,u0) by writing in an abstract operator form and. Of course, the unique so-lution of equations(l-3) will generate a continuous random RDS ψ(t,ω)u0=u=α(t,ω)-1u{t,ω;0, u0).In Chapter three: The random RDS ψ has a compact random attractor in H.Firstly, a random absorbing set B(0, K1(ω))for RDS ψ in H is got,Theorem3.1.1Assume v is a solution of (2.2.1)and (2.2.2),whereγ1>0,δ1>0and δ1γ1>|μ|2+|v|2,then there exists a random variable K1(Ω) satisfying the following property: For every p>0, there exists t(ω,p)≤-1, such that for any u(t0)∈H with|u(t0)|<ρ,and for any t0<t(ω,ρ), the followings estimates hold p-a.s|v(t,ω;t0,α(t0,ω)u0)|≤K1(Ω)(?)t∈[-1,0]. In particular,B(0, K1(Ω)) is a random absorbing set for RDS ψ in H.Then,a random absorbing setB(0, K3(ω))for RDS ψ in V is got,Theorem3.2.2Assume v is a solution of (2.2.1)and (2.2.2),whereγ1>0,δ1>0and δ1γ1>|μ|2+|v|2,then there exists one random variable K3(ω) satisfying the following properties. For all p>0there exists t(ω,ρ)≤-1such that the following hold p-a.s., for all t0≤t(ω,ρ) and u0∈H with‖u0‖<ρ,‖vx(0,ω;t0,α(t0,ω)u0)‖≤K3(ω). In particular, the ball B(0, K3(ω))is a random absorbing set in V.Thus,we obtain final conclusion.Theorem3.3.1The random dynamical system associated with the stochastic gener-alized Ginzburg-Landau equation (1) has a random attractor A(ω) namely, A(ω) is a compact,invariant set and attacts all bounded sets of L2(0,1), whereγ1>0,δ1>0and δ1γ1>|μ|2+|v|2.In Chapter four:More study about this equations are needed.