西南大学学报 (自然科学版)  2019, Vol. 41 Issue (9): 87-92.  DOI: 10.13718/j.cnki.xdzk.2019.09.011
0
Article Options
  • PDF
  • Abstract
  • Figures
  • References
  • 扩展功能
    Email Alert
    RSS
    本文作者相关文章
    王凤玲
    吴柯楠
    李扬荣
    欢迎关注西南大学期刊社
     

  • 非线性随机Ginzburg-Landau方程的Wong-Zakai逼近    [PDF全文]
    王凤玲, 吴柯楠, 李扬荣     
    西南大学 数学与统计学院, 重庆 400715
    摘要:主要证明了非线性随机Ginzburg-Landau方程在Wong-Zakai逼近意义下吸引子的存在性.
    关键词非线性随机Ginzburg-Landau方程    吸引子    Wong-Zakai逼近    

    目前已有很多文献从不同角度对Ginzburg-Landau方程进行了研究[1-3].本文参考文献[2],首先引入参数动力系统,定义了具有参数的方程所决定的动力过程Φ,针对这个动力过程,证明随机吸收集的存在性,再证明Φ的拉回渐进紧性,从而证明了在Wong-Zakai逼近意义下,非线性随机Ginzburg-Landau方程$\mathscr{D}$-拉回吸引子的存在性.

    1 预备知识

    本节参考文献[3],将引入具有参数的随机动力过程及其拉回吸引子的相关概念.设(Ω$\mathscr{F}$P)是一个随机空间,θt$\mathbb{R}$×ΩΩ是一个($\mathscr{B}$($\mathbb{R}$$\mathscr{F}$$\mathscr{F}$)可测映射,使得θt(0,·)是Ω上的恒等映射,且对任意的ts$\mathbb{R}$,满足θt(s+t,·)=θt(t,·)∘θt(s,·).

    定义1  令(Ω$\mathscr{F}$P,{θt}t$\mathbb{R}$)是参数动力系统,如果映射Φ$\mathbb{R}$+×$\mathbb{R}$×Ω×XX,对任意ωΩτ$\mathbb{R}$ts$\mathbb{R}$+,满足条件:

    (ⅰ) Φ(·,τ,·,·):$\mathbb{R}$+×Ω×XX是($\mathscr{B}$($\mathbb{R}$+$\mathscr{F}$×$\mathscr{B}$(X),$\mathscr{B}$(X))可测;

    (ⅱ) Φ(0,τω,·)是X上的恒等映射;

    (ⅲ) Φ(t+sτω,·)=Φ(tτ+sθsω,·)∘Φ(sτω,·);

    (ⅳ) Φ(tτω,·):XX是连续的,

    则称映射Φ是关于(Ω$\mathscr{F}$P,{θt}t$\mathbb{R}$)的连续动力过程.

    定义2  令$\mathscr{T}$X的所有有界非空子集族的集合,假设K={K(τω):τ$\mathbb{R}$ωΩ}∈$\mathscr{T}$.如果存在T=T(Dτω)>0,当tT时,对任意τ$\mathbb{R}$ωΩD$\mathscr{T}$,满足:

    $ \mathit{\Phi }\left( {t,\tau - t,{\theta _{ - t}}\omega ,D\left( {\tau - t,{\theta _{ - t}}\omega } \right)} \right) \subseteq K(\tau ,\omega ) $

    则称K为关于Φ$\mathscr{T}$-拉回吸收集.

    另外,如果∀τ$\mathbb{R}$ωΩK(τω)是X的非空闭子集,KΩ中关于$\mathscr{F}$可测,则称KΦ的闭可测$\mathscr{T}$-拉回吸收集.

    定义3  如果$\mathscr{A}$={$\mathscr{A}$(τω):τ$\mathbb{R}$ωΩ}∈$\mathscr{T}$,满足:

    (ⅰ) $\mathscr{A}$Ω中关于$\mathscr{F}$是可测的,并且对任意的ωΩ$\mathscr{A}$X中是紧的.

    (ⅱ) $\mathscr{A}$关于Φ是不变的.即对任意的t≥0,Φ(tτω$\mathscr{A}$(τω))=$\mathscr{A}$(τ+tθtω).

    (ⅲ) $\mathscr{A}$吸引$\mathscr{T}$中的每个元素:对于每个D$\mathscr{T}$

    $ \mathop {\lim }\limits_{t \to + \infty } {d_X}\left( {\mathit{\Phi }\left( {t,\tau - t,{\theta _{ - t}}\omega ,D\left( {\tau - t,{\theta _{ - t}}\omega } \right)} \right),\mathscr{A}(\tau ,\omega )} \right) = 0 $

    则称$\mathscr{A}$Φ$\mathscr{T}$-拉回吸引子.其中,$ {{d}_{X}}\left(A, B \right)=\underset{a\in A}{\mathop{\text{sup}}}\, \ \underset{b\in B}{\mathop{\text{inf}}}\, \|a-b{{\|}_{X}}$是Hausdorff-半距离.

    2 非线性随机Ginzburg-Landau方程的协循环

    $\mathscr{O}$$\mathbb{R}$n中的有界光滑区域,其中n=1,2.令τδ$\mathbb{R}$δ≠0.下面考虑如下的非线性Ginzburg-Landau方程:

    $ \left\{ {\begin{array}{*{20}{l}} {\frac{{\partial u}}{{\partial t}} - (\lambda + {\text{i}}\mu (t))\Delta u = \gamma u - (\kappa + {\text{i}}\beta (t))|u{|^2}u + g(t,x) + u{\mathscr{G}_\delta }\left( {{\theta _t}\omega } \right)} \\ {u(t,x) = 0\quad t \geqslant \tau ,x \in \partial \mathscr{O},u(\tau ,x) = {u_\tau }(x),x \in \mathscr{O}} \end{array}} \right. $ (1)

    其中未知量u是一个复值函数,λγκ>0,μ(·)∈$\mathscr{C}$($\mathbb{R}$$\mathbb{R}$),β(·)∈$\mathscr{C}$b($\mathbb{R}$$\mathbb{R}$),gLloc2($\mathbb{R}$L2($\mathscr{O}$)).

    δ≠0时,随机变量$\mathscr{G}$δ定义为:

    $ {\mathscr{G}_{\delta }}(\omega ) = \frac{{\omega (\delta )}}{\delta }\quad \forall \omega \in \mathit{\Omega } $ (2)

    存在一个θt不变量集Ω$\mathit{\widetilde{\Omega }}$,对于每个ω$\mathit{\widetilde{\Omega }}$,有

    $ \frac{{\omega (t)}}{t} \to 0\quad t \to \pm \infty $ (3)

    因此由(2)式可得

    $ {\mathscr{G}_{\delta }}\left( {{\theta _t}\omega } \right) = \frac{{\omega (t + \delta ) - \omega (t)}}{\delta }\quad \int_0^t {{\mathscr{G}_{\delta }}} \left( {{\theta _s}\omega } \right){\text{d}}s = \int_t^{t + \delta } {\frac{{\omega (s)}}{\delta }} {\text{d}}s + \int_\delta ^0 {\frac{{\omega (s)}}{\delta }} {\text{d}}s $ (4)

    由参考文献[3]、方程(4)以及ω的连续性可知,对于任意t$\mathbb{R}$,满足:

    $ \mathop {\lim }\limits_{\delta \to 0} \int_0^t {{\mathscr{G}_\delta }} \left( {{\theta _s}\omega } \right){\text{d}}s = \omega (t) - \omega (0) = \omega (t) $ (5)

    方程(1)是含有参数ωΩ的确定性方程.由文献[5]可知,如果(2)-(4)式满足,则对∀ωΩτ$\mathbb{R}$uτL2($\mathscr{O}$),方程(1)存在唯一的解u(·,τωuτ)∈C([τ,∞),L2($\mathscr{O}$))∩Lloc2([τ,∞),H01($\mathscr{O}$)).由此可以定义一个协循环Φ$\mathbb{R}$+×$\mathbb{R}$×Ω×L2($\mathscr{O}$)→L2($\mathscr{O}$),使得∀t$\mathbb{R}$+τ$\mathbb{R}$ωΩuτL2($\mathscr{O}$)满足

    $ \mathit{\Phi }\left( {t,\tau ,\omega ,{u_\tau }} \right) = u\left( {t + \tau ,\tau ,{\theta _{ - \tau }}\omega ,{u_\tau }} \right) $ (6)

    由定义1可知ΦL2($\mathscr{O}$)上关于(Ω$\mathscr{F}$P,{θt}t$\mathbb{R}$)的连续协循环.

    定义$\mathscr{D}$L2($\mathscr{O}$)的所有有界非空子集族的集合,即$\mathscr{D}$={D={D(τω):τ$\mathbb{R}$ωΩ}},∀c>0,τ$\mathbb{R}$ωΩ

    $ \mathop {\lim }\limits_{t \to - \infty } {{\text{e}}^{ct}}\left\| {D\left( {t + \tau ,{\theta _t}\omega } \right)} \right\| = 0 $ (7)

    其中

    $ \left\| D \right\| = \mathop {\sup }\limits_{u \in D} {\left\| u \right\|_{{L^2}(\mathscr{O})}} $

    D为缓增族.如果$\mathscr{D}$的所有元素为缓增的,则称$\mathscr{D}$为缓增的.

    本文中我们假设外力项g满足如下条件:存在常数α>0,使得

    $ \int_{ - \infty }^\tau {{{\text{e}}^{as}}} \left( {{{\left\| {g(s, \cdot )} \right\|}^2}} \right){\text{d}}s < \infty \quad \forall \tau \in \mathbb{R} $ (8)

    当证明缓增拉回吸收集存在时,需要假设:存在常数α>0,对任意c>0,

    $ \mathop {\lim }\limits_{t \to - \infty } {{\text{e}}^{ct}}\int_{ - \infty }^0 {{{\text{e}}^{as}}} \left( {{{\left\| {g(s + t, \cdot )} \right\|}^2}} \right){\text{d}}s = 0 $ (9)
    3 解的一致性估计

    本节将对方程(1)的解在L2($\mathscr{O}$)空间上进行一致性估计.

    引理1  对于τ$\mathbb{R}$ωΩ以及D={D(τω):τ$\mathbb{R}$ωΩ}∈$\mathscr{D}$,存在T=T(τωD)>0,使得对于∀tT,方程(1)中的u满足:

    $ \begin{array}{*{20}{l}} {{{\left\| {u\left( {\tau ,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|}^2} + 2\lambda \int_{\tau - t}^\tau {{{\rm{e}}^{a(s - \tau )}}} {{\left\| {\nabla u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|}^2}{\rm{d}}s \le }\\ {M\int_{ - \infty }^0 {{{\rm{e}}^{as}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2} + {{\left\| {g(s + \tau )} \right\|}^2}} \right){\rm{d}}s} \end{array} $ (10)
    $ \begin{array}{l} \int_{\tau - t}^\tau {{{\rm{e}}^{\alpha (s - \tau )}}} \left( {{{\left\| {\nabla u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|}^2} + \left\| {u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|_{L^4}^4} \right) \le \\ M\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2} + {{\left\| {g(s + \tau )} \right\|}^2}} \right){\rm{d}}s \end{array} $ (11)

      将方程(1)与u的共轭u$\mathscr{O}$上作内积并且取实部,可得

    $ \frac{1}{2}\frac{\partial }{{\partial t}}{\left\| u \right\|^2} + \lambda {\left\| {\nabla u} \right\|^2} = \gamma {\left\| u \right\|^2} + \text{Re} ((g(t,x),u(t,x))) - \kappa \left\| u \right\|_{{L^4}}^4 + {\mathscr{G}_\delta }\left( {{\theta _t}\omega } \right){\left\| u \right\|^2} $ (12)

    对(12)式右边第二项估计,可得

    $ \left| {(g(t,x),u(t,x))} \right| \le \frac{\alpha }{2}{\left\| u \right\|^2} + \frac{1}{{2\alpha }}{\left\| g \right\|^2} $ (13)

    对(12)式右边第三项和第四项估计,可得

    $ \kappa \left\| u \right\|_{{L^4}}^4 - {\mathscr{G}_\delta }\left( {{\theta _t}\omega } \right){\left\| u \right\|^2} \ge - \frac{{{{\left| {{\mathscr{G}_\delta }\left( {{\theta _t}\omega } \right)} \right|}^2}}}{{4\kappa }}\left| \mathscr{O} \right| $ (14)

    由(13),(14)式可知,

    $ \begin{array}{l} \begin{array}{*{20}{l}} {\frac{\partial }{{\partial t}}{{\left\| u \right\|}^2} + 2\lambda {{\left\| {\nabla u} \right\|}^2} + \kappa \left\| u \right\|_{{L^4}}^4 \le }\\ {2\gamma {{\left\| u \right\|}^2} + \alpha {{\left\| u \right\|}^2} + \frac{1}{\alpha }{{\left\| g \right\|}^2} - \kappa \left\| u \right\|_{{L^4}}^4 + 2{\mathscr{G}_\delta }\left( {{\theta _t}\omega } \right){{\left\| u \right\|}^2} = } \end{array}\\ - \kappa \left\| u \right\|_{{L^4}}^4 + 2\left( {{\mathscr{G}_\delta }\left( {{\theta _t}\omega } \right) + (\gamma + \alpha )} \right){\left\| u \right\|^2} + \frac{1}{\alpha }{\left\| g \right\|^2} - \alpha {\left\| u \right\|^2} \le \\ \frac{{{{\left| {{\mathscr{G}_\delta }\left( {{\theta _t}\omega } \right) + (\gamma + \alpha )} \right|}^2}}}{\kappa }\left| \mathscr{O} \right| + \frac{1}{\alpha }{\left\| g \right\|^2} - \alpha {\left\| u \right\|^2} \end{array} $ (15)

    综上可得

    $ \frac{\partial }{{\partial t}}{\left\| u \right\|^2} + 2\lambda {\left\| {\nabla u} \right\|^2} + \kappa \left\| u \right\|_{{L^4}}^4 + \alpha {\left\| u \right\|^2} \le \frac{{{{\left| {{\mathscr{G}_\delta } + (\gamma + \alpha )} \right|}^2}}}{\kappa }\left| \mathscr{O} \right| + \frac{1}{\alpha }{\left\| g \right\|^2} $ (16)

    在(16)式两边乘以eαt,在(τ-tτ)上积分,其中τrωΩ,令θ-τω替代ω可得

    $ \begin{array}{l} {\left\| {u\left( {\tau ,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|^2} + 2\lambda \int_{\tau - t}^\tau {{{\rm{e}}^{\alpha (s - \tau )}}} {\left\| {\nabla u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|^2}{\rm{d}}s \le \\ {{\rm{e}}^{ - \alpha t}}{\left\| {{u_{\tau - t}}} \right\|^2} + \int_{\tau - t}^\tau {{{\rm{e}}^{\alpha (s - \tau )}}} \left( {\frac{1}{\alpha }{{\left\| {g(s)} \right\|}^2}} \right){\rm{d}}s + \frac{{{C_0}}}{\kappa }\int_{\tau - t}^\tau {{{\rm{e}}^{\alpha (s - \tau )}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _{s - \tau }}\omega } \right)} \right| +\\ {{\left| {{\mathscr{G}_\delta }\left( {{\theta _{s - \tau }}\omega } \right)} \right|}^2}} \right){\rm{d}}s \le \\ {{\rm{e}}^{ - \alpha t}}{\left\| {{u_{\tau - t}}} \right\|^2} + \int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}\left( {\frac{1}{\alpha }{{\left\| {g(s + \tau )} \right\|}^2}} \right){\rm{d}}s} + \frac{{{C_0}}}{\kappa }\int_{\tau - t}^\tau {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2}} \right){\rm{d}}s \end{array} $ (17)

    由(4),(5),(7)式可知,对于uτ-tD(τ-tθ-tω),D$\mathscr{D}$,当t→∞时有

    $ {{\rm{e}}^{ - \alpha t}}{\left\| {{u_{\tau - t}}} \right\|^2} \le {{\rm{e}}^{ - \alpha t}}\left\| {D\left( {\tau - t,{\theta _{ - \tau }}\omega } \right)} \right\| \to 0 $

    故存在T=T(τωD)>0,使得对∀tT

    $ {{\rm{e}}^{ - \alpha t}}{\left\| {{u_{\tau - t}}} \right\|^2} \le \int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2}} \right){\rm{d}}s $ (18)

    结合(17),(18)式,可知(10)式成立.

    另一方面,结合(17),(18)式,对(16)式在(τ-tτ)上运用Gronwall不等式,并令θ-τω替代ω可得

    $ \begin{array}{*{20}{l}} {\int_{\tau - t}^\infty {{{\rm{e}}^{\alpha (s - \tau )}}} \left( {{{\left\| {\nabla u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|}^2} + \left\| {u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|_{{L^4}}^4} \right) \le }\\ {M\int_{ - \infty }^0 {{{\rm{e}}^{as}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2} + {{\left\| {g(s + \tau )} \right\|}^2}} \right){\rm{d}}s} \end{array} $ (19)

    则引理1得证.

    推论1  假设(8),(9)式成立,则连续协循环Φ具有一个闭的、可测的$\mathscr{D}$-拉回吸引集K={K(τω):τ$\mathbb{R}$ωΩ}∈$\mathscr{D}$.对∀τ$\mathbb{R}$以及ωΩK定义为

    $ K(\tau ,\omega ) = \left\{ {u \in {L^2}(\mathscr{O}):{{\left\| u \right\|}^2} \le {R_1}(\tau ,\omega )} \right\} $

    其中

    $ {R_1}(\tau ,\omega ) = M\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2} + {{\left\| {g(s + \tau )} \right\|}^2}} \right){\rm{d}}s $ (20)

      对给定的τ$\mathbb{R}$ωΩD$\mathscr{D}$,由定义2和引理1知,存在T=T(τωD)>0,使得对任意tT

    $ \mathit{\Phi }\left( {t,\tau - t,{\theta _{ - t}}\omega ,D\left( {\tau - t,{\theta _{ - t}}\omega } \right)} \right) = u\left( {\tau ,\tau - t,{\theta _{ - \tau }}\omega ,D\left( {\tau - t,{\theta _{ - t}}\omega } \right)} \right) \subseteq K(\tau ,\omega ) $ (21)

    P是任意正数,

    $ \begin{array}{l} \mathop {\lim }\limits_{t \to - \infty } {{\rm{e}}^{Pt}}{\left\| {K\left( {\tau + t,{\theta _t}\omega } \right)} \right\|^2} = \mathop {\lim }\limits_{t \to - \infty } {{\rm{e}}^{Pt}}{R_1}\left( {\tau + t,{\theta _t}\omega } \right) = \\ \mathop {\lim }\limits_{t \to - \infty } M{{\rm{e}}^{Pt}}\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _{s + t}}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _{s + t}}\omega } \right)} \right|}^2}} \right){\rm{d}}s + \\ \mathop {\lim }\limits_{t \to - \infty } M{{\rm{e}}^{Pt}}\int_{ - \infty }^0 {{{\rm{e}}^{as}}} \left( {{{\left\| {g(s + \tau + t)} \right\|}^2}} \right){\rm{d}}s \end{array} $ (22)

    对(22)式右边第一项估计,令N=min{Pα},对任意的t≤0,可得

    $ \begin{array}{*{20}{l}} {M{{\rm{e}}^{Pt}}\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _{s + t}}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _{s + t}}\omega } \right)} \right|}^2}} \right){\rm{d}}s \le }\\ {M\int_{ - \infty }^0 {{{\rm{e}}^{N(s + t)}}} \left( {\left| {{{\cal G}_\delta }\left( {{\theta _{s + t}}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _{s + t}}\omega } \right)} \right|}^2}} \right){\rm{d}}s \le }\\ {M\int_{ - \infty }^t {{{\rm{e}}^{Ns}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2}} \right){\rm{d}}s} \end{array} $ (23)

    通过(3),(4)式,可得

    $ \int_{ - \infty }^0 {{{\rm{e}}^{Ns}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2}} \right){\rm{d}}s < \infty $

    因此

    $ \mathop {\lim }\limits_{t \to - \infty } M{{\rm{e}}^{Pt}}\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2}} \right){\rm{d}}s < 0 $ (24)

    结合(10)式,对(22)式右边第二项估计,可得

    $ \mathop {\lim }\limits_{t \to - \infty } M{{\rm{e}}^{Pt}}\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {{{\left\| {g(s + \tau + t)} \right\|}^2}} \right){\rm{d}}s = M{{\rm{e}}^{ - P\tau }}\mathop {\lim }\limits_{t \to - \infty } {{\rm{e}}^{Pt}}\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {{{\left\| {g(s + t)} \right\|}^2}} \right){\rm{d}}s = 0 $ (25)

    结合(24),(25)式,对于任意的P>0,可得

    $ \mathop {\lim }\limits_{t \to - \infty } {{\rm{e}}^{Pt}}{\left\| {K\left( {\tau + t,{\theta _t}\omega } \right)} \right\|^2} = 0 $ (26)

    因此K(τω)在空间L2($\mathscr{O}$)上是缓增的.另外,对∀τ$\mathbb{R}$L(τ,·):Ω$\mathbb{R}$是可测的.

    引理2  存在随机半径R2(τω):假设(8),(9)式成立,对∀τ$\mathbb{R}$ωΩD={D(τω):τ$\mathbb{R}$ωΩ}∈$\mathscr{D}$,存在T=T(τωD)>0,使得对∀tT,满足

    $ \int_{\tau - 1}^\tau {{{\left\| {\nabla u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|}^2}} {\rm{d}}s \le {R_2}(\tau ,\omega ) $ (27)

      由引理1,存在T=T(τωD)≥1,对∀tT

    $ \begin{array}{l} \lambda \int_{\tau - 1}^\tau {{{\rm{e}}^{\alpha s}}} {\left\| {\nabla u\left( {\tau ,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|^2}{\rm{d}}s \le \\ \lambda \int_{\tau - t}^\tau {{{\rm{e}}^{\alpha s}}} {\left\| {\nabla u\left( {\tau ,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|^2}{\rm{d}}s \le \\ M\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2} + {{\left\| {g(s + \tau )} \right\|}^2}} \right){\rm{d}}s \end{array} $ (28)

    $ {R_2}(\tau ,\omega ) = \frac{{{R_1}(\tau ,\omega )}}{\lambda } = \frac{M}{\lambda }\int_{ - \infty }^0 {{{\rm{e}}^{\alpha s}}} \left( {\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right| + {{\left| {{\mathscr{G}_\delta }\left( {{\theta _s}\omega } \right)} \right|}^2} + {{\left\| {g(s + \tau )} \right\|}^2}} \right){\rm{d}}s $

    根据eαs≥eατe-α,对所有的sτ-1,引理2成立.

    引理3  存在随机半径R3(τω):对于τ$\mathbb{R}$ωΩ以及D={D(τω):τ$\mathbb{R}$ωΩ}∈$\mathscr{D}$,存在T=T(τωD)>0,使得对于∀tT$\exists $ε>0,σ$\mathbb{R}$,满足

    $ {\left\| {\nabla u\left( {\tau ,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|^2} \le {R_3}(\tau ,\omega ) $ (29)

      对于所有的tT,将方程(1)与-Δu$\mathscr{O}$上作内积并且取实部,可得

    $ \begin{array}{*{20}{c}} {\frac{1}{2}\frac{\partial }{{\partial t}}{{\left\| {\nabla u} \right\|}^2} + \lambda {{\left\| {\Delta u} \right\|}^2} = \gamma {{\left\| {\nabla u} \right\|}^2} + }\\ {\left. {\text{Re} ((\kappa + {\rm{i}}\beta (t)))\left( {{{\left| u \right|}^2}u,\Delta u} \right)} \right) - \text{Re} (g(t, \cdot ),\Delta u) + {\mathscr{G}_\delta }(\theta ,\omega ){{\left\| {\nabla u} \right\|}^2}} \end{array} $ (30)

    根据Young不等式可知

    $ \left| {(g(t,x),\Delta u(t,x))} \right| \le \frac{\lambda }{4}{\left\| {\Delta u} \right\|^2} + C{\left\| {g(t, \cdot )} \right\|^2} $

    再根据Sobolev嵌入定理和内插不等式可知,对任意的uH10($\mathscr{O}$),

    $ \left\| {\nabla u} \right\|_4^2 \le C\left\| {\nabla u} \right\|{\left( {{{\left\| u \right\|}^2} + {{\left\| {\Delta u} \right\|}^2}} \right)^{\frac{1}{2}}} $

    则对(30)式右边第二项估计,可得

    $ \text{Re} \left( {(\kappa + {\rm{i}}\beta (t))\left( {{{\left| u \right|}^2}u,\Delta u} \right)} \right) \le \frac{\lambda }{4}\left( {{{\left\| u \right\|}^2} + {{\left\| {\Delta u} \right\|}^2}} \right) + C\left\| u \right\|_{{L^4}}^4{\left\| {\nabla u} \right\|^2} $ (31)

    综上可知

    $ \frac{\partial }{{\partial t}}{\left\| {\nabla u} \right\|^2} \le C\left( {1 + \left\| u \right\|_{{L^4}}^4 + {\mathscr{G}_\delta }\left( {{\theta _t}\omega } \right)} \right){\left\| {\nabla u} \right\|^2} + C{\left\| u \right\|^2} + C{\left\| {g(t, \cdot )} \right\|^2} $ (32)

    任给t≥0,τ$\mathbb{R}$ωΩ,取s∈(τ-1,τ),在区间(τ-1,τ)上运用Gronwall不等式,并将ω替换成θ-τω,可得

    $ \begin{array}{l} \begin{array}{*{20}{l}} {{{\left\| {\nabla u\left( {\tau ,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|}^2} \le C{{\rm{e}}^{\int_{\tau - 1}^\tau {\left( {1 + \left\| {u\left( {s,\tau - t,{\theta _{ - \tau }},{u_{\tau - t}}} \right)} \right\|_4^4 + {\mathscr{G}_\delta }\left( {{\theta _s},\omega } \right)} \right)} }}{\rm{d}}s}\\ {\left( {C\int_{\tau - 1}^\tau {\left( {{{\left\| {u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|}^2} + {{\left\| {g(s, \cdot )} \right\|}^2}} \right)} {\rm{d}}s + } \right.} \end{array}\\ \left. {\int_{\tau - 1}^\tau {{{\left\| {\nabla u\left( {s,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|}^2}{\rm{d}}s} } \right) \end{array} $ (33)

    其中,结合(3),(4)式以及文献[4]可知,存在ε>0,σ$\mathbb{R}$h<0,使得$|\int_{0}^{0}{{{\mathscr{G}}_{\delta }}({{\theta }_{\sigma }}\omega)\text{d}\sigma }|\le -\varepsilon h+{{C}_{\sigma }}(\omega)$成立.

    由引理1、引理2和(9)式可得

    $ {\left\| {\nabla u\left( {\tau ,\tau - t,{\theta _{ - \tau }}\omega ,{u_{\tau - t}}} \right)} \right\|^2} \le C\left( {{R_1}(\tau ,\omega ) + {R_2}(\tau ,\omega )} \right){{\rm{e}}^{C\left( {1 + {R_1}\left( {\tau ,\omega } \right) + {C_\sigma }\left( \omega \right)} \right)}} $

    成立.令

    $ C\left( {{R_1}(\tau ,\omega ) + {R_2}(\tau ,\omega )} \right){{\rm{e}}^{C\left( {1 + {R_1}(\tau ,\omega ) + {C_\sigma }(\omega )} \right)}} = {R_3}(\tau ,\omega ) $

    从而引理3成立.

    4 主要结论

    定理1  假设(8),(9)式成立,则协循环ΦL2($\mathscr{O}$)上有唯一的$\mathscr{D}$-拉回吸引子$\mathscr{A}$={$\mathscr{A}$(τω):τ$\mathbb{R}$ωΩ}∈$\mathscr{D}$.

      由引理1可知,Φ有一闭的、可测的$\mathscr{D}$-拉回吸收集K,由引理3可知,ΦL2($\mathscr{O}$)上有一个紧的吸收集.因此,由文献[5]吸引子的存在性结论,可得协循环Φ存在$\mathscr{D}$-拉回吸引子.

    参考文献
    [1]
    王蕊, 李扬荣. 带有可乘白噪音的广义Ginzburg-Landau方程的随机吸引子[J]. 西南大学学报(自然科学版), 2012, 34(2): 92-95.
    [2]
    LU K, WANG B. Wong-Zakai Approximations and Long Term Behavior of Stochastic Partial Differential Equations[J]. Journal of Dynamics Differential Equations, 2017(4): 1-31.
    [3]
    WANG B. Sufficient and Necessary Criteria for Existence of Pullback Attractors for Non-Compact Random Dynamical Systems[J]. Journal of Differential Equations, 2012, 253(5): 1544-1583. DOI:10.1016/j.jde.2012.05.015
    [4]
    WANG B. Random Attractors for Non-Autonomous Stochastic Wave Equations with Multiplicative Noise[J]. Discrete Continuous Dynamical Systems-Series A (DCDS-A), 2013, 34(1): 269-300. DOI:10.3934/dcds.2014.34.269
    [5]
    TEMAM R. Infinite-Dimensional Dynamical Systems in Mechanics and Physics[M]. New York: Springer-Verlag, 1997.
    Wong-Zakai Approximations of Nonlinear Stochastic Ginzburg-Landau Equations
    WANG Feng-ling, WU Ke-nan, LI Yang-rong     
    School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
    Abstract: In this paper, we mainly prove, through the Wong-Zakai approximations, the existence of pullback random attractors of nonlinear stochastic Ginzburg-Landau equations.
    Key words: nonlinear stochastic Ginzburg-Landau equation    Pullback random attractor    Wong-Zakai approximation    
    X