该文研究了下列带加性噪声的非局部扩散方程的解的渐近行为

${\label{e1.1}} \left\{ \begin{array}{l} d({u}-\Delta {u}) - ( {\Delta {u} - \lambda{u}} ){\rm d}t = ( {f (x,{u} ) +g ( x )} ){\rm d}t +{h ( {x} ){\rm d}w} ,x\in {\Bbb R}^n,\,\,t > 0,\\ {u}(0,x) ={u}_0 ,x\in {\Bbb R}^n,\\ \end{array} \right.$

(1.1)

其中 $\lambda$为一正数,$g\in L^2({\Bbb R}^n)$,$h\in H^2({\Bbb R}^n)\cap W^{2,p}({\Bbb R}^n)$ 对 $p\geq 2$,$f$ 为满足某增长条件的非线性函数,$w$ 是双边实值Wiener过程.

非局部反应扩散方程(1.1)常出现在流体力学,固体力学和热传导理论中^[1-3]. 在确定性情形 ($h=0$),方程 (1.1)的解的适定性和吸引子的存在性已经有许多研究^[4-10]. 然而,据作者所知,即使是有界情形,对随机非局部反应扩散方程的吸引子的研究都很少. 该文将研究这类问题.

拉回意义的随机吸引子概念首先由Crauel,Flandoli等引入,参见文献[11-14],随后,随机吸引子得到蓬勃发展,许多学者相继研究了各类方程吸引子存在性问题,参见文献[16-27]. 尤其,Bates等^[15]首先应用尾估计的方法克服了无界区域Sobolev 嵌入不紧的困难而建立了定义在无界区域上的随机反应扩散方程产生的随机动力系统的拉回渐近紧性并获得了随机吸引子存在的充分条件. 由于方程(1.1)含有 $-\Delta_t$项,这使得其和通常的反应扩散方程有本质区别. 例如： 通常的反应扩散方程有某种正则性,即虽然初值属于弱拓扑空间,但方程的解属于更强的拓扑空间. 然则,方程(1.1)没有正则性. 这给我们建立非局部扩散方程随机吸引子存在性增加了难度. 为了克服无界区域Sobolev嵌入不紧和没有正则性的问题,该文运用尾估计和分解相结合的方法证明方程解的渐近紧性.

该部分将证明下列定义在${\Bbb R}^n$上的带加性白噪声的非局部扩散方程生成连续随机动力系统

${\label{e3.1}} \left\{ \begin{array}{ll} d({u}-\Delta {u}) - ( {\Delta {u} - \lambda{u}} ){\rm d}t = ( {f (x,{u} ) +g(x)} ){\rm d}t +{h ( {x} ){\rm d}w} ,& x\in {\Bbb R}^n,\,\,t > 0,\\ {u}(0,x) ={u}_0 ,& x\in {\Bbb R}^n,\end{array} \right.$

(2.1)

其中 $\lambda$为一正数,$g\in L^2({\Bbb R}^n)$,$h\in H^2({\Bbb R}^n)\cap W^{2,p}({\Bbb R}^n)$ 对 $p\geq 2$,$w$ 是双边实值Wiener过程,$f$ 为满足下列条件的非线性函数

${f ( {x,s} )s} \le -\alpha _1 \left| s \right|^{p} + \psi _1 (x),\label{c3.1}$

(2.2)

$\left| {f( {x,s} )} \right| \le \alpha _2 \left| s \right|^{p-1} + \psi _2 (x),\label{c3.2}$

(2.3)

其中$\alpha _1$ 和 $\alpha _2$为正常数,当$n=1,2$时,$2\leq p＜\infty$; 当$n\geq3$时,$2\leq p＜ \frac{2n}{n-2}$; $\psi _1\in L^1({\Bbb R}^n)$; $\psi _2\in L^2({\Bbb R}^n)$.

下面,考虑概率空间 $(\Omega ,{\cal F},P)$,其中

$\Omega = \left\{ {\omega \in C ( {\Bbb R},{\Bbb R } ):\omega ( 0 ) = 0} \right\},$

${\cal F}$ 为由$\Omega $诱导的紧开拓扑 Borel $\sigma$ -代数,$P$ 为$(\Omega ,{\cal F})$上的相应的 Wiener 测度. 定义时间转移为

$\theta _t \omega ( \cdot) = \omega ( { \cdot + t} ) - \omega ( t ),\,\,\omega \in \Omega ,\,\,t \in {\Bbb R}. $

那么 $(\Omega ,{\cal F},P,(\theta_t)_{t\in {\Bbb R}})$为一度量动力系统.

接下来,证明随机方程(2.1)生成连续随机动力系统. 为此,先将带白噪声的随机方程转化为带随机参数的方程.

考虑一维Ornstein-Uhlenbeck方程

${\label{3.2}} {\rm d}y + \lambda y{\rm d}t = {\rm d}w ( t ).$

(2.4)

易知方程(2.4)的解为

$y (t) = y( {\theta _t \omega }) \equiv - \lambda \int_{ - \infty }^0 {{\rm e}^{\lambda \tau } ( {\theta _t \omega} )( \tau )} {\rm d}\tau ,t \in {\Bbb R}. $

由文献[28]可知,存在$\theta_{t}$ -不变集 $\widetilde \Omega \subseteq \Omega $,关于$P$ 保测,对任意$\omega\in \widetilde \Omega $,$| y(\theta_t\omega) |$ 关于$t$连续,并且随机变量$| y(\omega) |$是缓增的. 设${\cal F}_1$ 和 $P_1$ 为 $\mathcal F$ 和 $P$ 在 $\widetilde \Omega $ 上的限制,我们将定义问题(2.1)在 $(\widetilde\Omega ,{\cal F}_1,P_1,(\theta_{t})_{t\in R})$ 上的连续随机动力系统. 为方便起见,从今以后,我们仍然将$(\widetilde \Omega ,{\cal F}_1,P_1)$写成 $( \Omega ,{\cal F},P)$. 因此,综上所述,存在一缓增函数$r(\omega) > 0$,以致对每个$\omega\in \Omega $,有

${\label{3.3}} ( | y (\omega )|^2 + | y(\omega ) |^p ) \le r(\omega ).$

(2.5)

设$z ( {\theta _t \omega }) = (I-\Delta)^{-1}hy(\theta_t\omega),$ 由方程(2.4) 可得

$d(z-\Delta z) + \lambda (z-\Delta z){\rm d}t = h{\rm d}w (t). $

令$v(t) = u(t)- z(\theta_t\omega)$,其中$u$是方程 (2.1)的解. 那么$v$满足

$\left\{ \begin{array}{ll} v_t-\Delta v_t - \Delta v +\lambda v = {f(x,{v + z(\theta_t\omega )} ) + g (x)} + (1-\lambda)\Delta z(\theta_t\omega ),&x \in {\Bbb R}^n ,\,t > 0,\\ v(0,x)= v_0(x)=u_0(x)- z(\omega),&x \in {\Bbb R}^n. \end{array} \right.$

(2.6)

由Galerkin方法可知如果$f$满足条件(2.2)-(2.3),那么对每一个$\omega\in \Omega $和对任意$v_0\in H^1({\Bbb R}^n)$,方程(2.6) 有唯一解$v( { \cdot ,\omega ,v_0 } ) \in C( {\left[{0,\infty} \right),H^1({\Bbb R}^n) } ) \cap L^{p} ( {( {0,T} ),L^{p} ({\Bbb R}^n) } ) $,对任意$T>0$,其中$v( {0,\omega ,v_0 } ) = v_0$. 进一步,对任意$t \geq 0$,$v(t,\omega,v_0)$关于 $v_0\in H^1({\Bbb R}^n)$在$H^1({\Bbb R}^n)$上是连续的. 令 $u( {t,\omega ,u_0 } ) = v( {t,\omega ,u_0 - z(\omega )} ) + z(\theta_t\omega ). $ 则$u$是方程(2.1)的解. 定义一映射$\phi : {\Bbb R}^+ \times \Omega \times H^1({\Bbb R}^n) \to H^1({\Bbb R}^n) $为

$\phi( {t,\omega ,u_0 } ) = u( {t,\omega ,u_0 } ) = v( {t,\omega ,u_0 - z ( \omega )} ) + z(\theta_t\omega ),~~(t,\omega ,u_0)\in {\Bbb R}^+ \times \Omega \times H^1({\Bbb R}^n).$

(2.7)

由文献[15]可知,$\phi$为连续随机动力系统.

定义

$\sigma=\min\Big\{\frac{\lambda}{4},\frac{1}{4}\Big\}.$

(2.8)

下列,都假设${\cal D}$ 为关于$(\Omega ,{\cal F},P,(\theta_t)_{t\in {\Bbb R}})$的$H^1({\Bbb R}^n )$中的缓增集. 该部分主要是给出问题(2.6)解的一系列估计.

引理3.1 假设条件(2.2)-(2.3)成立. 设 $\{B(\omega)\}_{\omega\in \Omega } \in {\cal D}$ 和 $v_0(\omega)\in B(\omega)$. 那么对每一个 $\omega \in \Omega $,存在$T=T(B,\omega)>0$,以致方程(2.6)的解$v(t,\omega,v_0({\omega}))$满足对任意 $t\geq T$,

$\left\| {v ( {t,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )} \le M({1+ r(\omega )}) ,$

(3.1)

其中$M$是一依赖于$\lambda$ 的正常数,但是不依赖于$\omega$ 和 $B$,$r(\omega)$是一缓增函数.

证 方程(2.6) 关于 $v$ 在$L^2({\Bbb R}^n)$中取内积,可得

$ \frac{1}{2}\frac{\rm d}{{\rm d}t}(\left\| {v } \right\|^2+\left\|\nabla {v } \right\|^2) + \left\| {\nabla v } \right\|^2 +\lambda \left\| {v } \right\|^2 \\ = ( {f( {v +z( {\theta _t \omega } )} ),v } ) + ( {g (x),v } ) +(1-\lambda) ( {\Delta z(\theta_t\omega ),v } ).$

(3.2)

对于非线性项,由条件(2.2)-(2.3) 可得

$( {f(x,{v + z(\theta_t\omega )} ),v } ) = \int_{{\Bbb R}^n} f(x,{v + z(\theta_t\omega )} )v {\rm d}x \\ = \int_{{\Bbb R}^n} f(x,{v + z(\theta_t\omega )} )( {v + z(\theta_t\omega )} ){\rm d}x \\ \mbox{}- \int_{{\Bbb R}^n} f(x,{v + z( {\theta _t \omega } )} )z(\theta_t\omega ){\rm d}x \\ \le - \alpha _1 \int_{{\Bbb R}^n } {\left| u \right|^p } {\rm d}x + \int_{{\Bbb R}^n } {\psi _1 ( x )} {\rm d}x \\ \mbox{}+ \alpha _2 \int_{{\Bbb R}^n } {\left| u \right|^{p - 1} \left| {z ( {\theta _t \omega } )} \right|} {\rm d}x + \int_{{\Bbb R}^n } {\left| {\psi _2 (x)} \right|\left| {z(\theta_t\omega )} \right|} {\rm d}x\\ \leq - \frac{1}{2}\alpha _1 \left\| {u } \right\|_{p}^{p} + c( {\left\| {z(\theta_t\omega )} \right\|_{p}^{p} + \left\| {z(\theta_t\omega )} \right\|^2 } ) + c.$

(3.3)

另一方面,不等式 (3.2)的右边的最后两项有如下估计

$( {g (x),v } ) +(1-\lambda) ( {\Delta z(\theta_t\omega ),v } ) \le \frac{1}{2}\lambda \left\| {v } \right\|^2 + \frac{1}{2\lambda } \left\| {g (x)} \right\|^2 + c\left\| {\nabla z(\theta_t\omega )} \right\|^2 +\frac{1}{2} \left\|\nabla {v } \right\|^2 .$

(3.4)

那么,由 (3.2)-(3.4)式得

$ \frac{\rm d}{{\rm d}t}(\left\| {v } \right\|^2 +\left\|\nabla {v } \right\|^2 )+\left\| \nabla{v } \right\|^2 + \lambda \left\| {v} \right\|^2 +\alpha _1 \left\| {u } \right\|_p^p \\ \leq c( {\left\| {z(\theta_t\omega )} \right\|_p^p + \left\| {z(\theta_t\omega )} \right\|^2 + \left\| {\nabla z( {\theta _t \omega } )} \right\|^2 } ) + c.$

(3.5)

注意 $(I-\Delta)^{-1}h\in H^2({\Bbb R}^n)\cap W^{2,p}({\Bbb R}^n)$. 故(3.5)式的右边可以如下估计

$c{( {| {y ( {\theta _t \omega } )} |^p + | {y( {\theta _t \omega} )} |^2 } )} + c = p_1 (\theta_t\omega ) + c.$

(3.6)

由 (2.5)式和 $r(\omega)$的缓增性,可知对每一个 $\omega\in \Omega $,

$p_1 (\theta_t\omega ) \le c{\rm e}^{\frac{1}{2} \sigma \left| t \right|} r(\omega ),\forall t \in {\Bbb R}.$

(3.7)

由 (3.5)-(3.6) 式和 (2.8)式可知,对任意 $t\geq 0$,

$\frac{\rm d}{{\rm d}t}(\left\| {v } \right\|^2+\left\|\nabla {v } \right\|^2) +2 \sigma (\left\| {v } \right\|^2 + \left\| {\nabla v } \right\|^2 )+ \left\| {u } \right\|_p^p \le p_1 (\theta_t\omega ) + c.$

(3.8)

对上式两边先乘于 ${\rm e}^{\sigma t}$,再在 $(0,t)$上积分可知,对任意 $t\geq0$,

$ \left\| {v ( {t,\omega ,v_0( \omega )} )} \right\|^2_{H^1({\Bbb R}^n)} + \int_0^t {{\rm e}^{\sigma ( {\tau - t} )} } \left\| {u ( {\tau ,\omega ,u_0 (\omega )} )} \right\|_p^p {\rm d}\tau \\ \mbox{} +\sigma \int_0^t {{\rm e}^{\sigma ( {\tau - t} )} } \left\| {v ( {\tau,\omega ,v_0( \omega )} )} \right\|^2_{H^1({\Bbb R}^n)} {\rm d}\tau \\ \le {\rm e}^{ - \sigma t} \left\| {v_0 (\omega )} \right\|^2_{H^1({\Bbb R}^n)} + \int_0^t {{\rm e}^{\sigma ( {\tau - t} )} } p_1 ( {\theta _\tau \omega } ){\rm d}\tau + \frac{c}{\sigma}.$

(3.9)

用$\theta_{-t}\omega$替换$\omega$,由(3.7)和 (3.9)式可得,对任意 $t\geq0$,

$\left\| {v ( {t,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1({\Bbb R}^n) }^2 +\sigma \int_0^t {{\rm e}^{\sigma ( {\tau - t} )} } \left\| {v ( {\tau,\theta _{ - t}\omega ,v_0( \theta _{ - t}\omega )} )} \right\|^2_{H^1({\Bbb R}^n)}\\ \le {\rm e}^{ - \sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 + \int_0^t {{\rm e}^{\sigma( {s - t} )} } p_1 ( {\theta _{s - t} \omega } ){\rm d}s + \frac{c}{\sigma } \\ \le {\rm e}^{ - \sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 + \int_{ - t}^0 {{\rm e}^{\sigma\tau } } p_1 ( {\theta _\tau \omega } ){\rm d}\tau + \frac{c}{\sigma} \\ \le {\rm e}^{ - \sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 + c\int_{ - t}^0 {{\rm e}^{\frac{1}{2}\sigma \tau } } r(\omega ){\rm d}\tau + \frac{c}{\sigma } \\ \le {\rm e}^{ - \sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 + \frac{{2c}}{\sigma }r(\omega ) + \frac{c}{\sigma}.$

(3.10)

由假设,$\{B(\omega)\}_{\omega\in \Omega } \in {\cal D}$ 是缓增的. 因此,如果 $ v_0 ( {\theta _{ - t} \omega } ) \in B( {\theta _{ - t} \omega } ) $,那么存在$T=T(B,\omega)>0$,对任意$t\geq T$,

${\rm e}^{ - \sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 \le cr(\omega ) + c. $

上式结合(3.10) 式可得结论.

引理3.2 假设条件(2.2)-(2.3)成立. 设 $\{B(\omega)\}_{\omega\in \Omega } \in {\cal D}$ 和 $v_0(\omega)\in B(\omega)$. 那么对每一个 $\omega \in \Omega $ 和每一个 $T_1>0$,以致(2.6)式的解$v(t,\omega,v_0({\omega}))$满足对任意 $t\geq T_1$,

$\int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} \left\| {v ( {s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1({\Bbb R}^n) }^2 }{\rm d}s \le {\rm e}^{ - \sigma t} \left\| {v_0( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 + M( {1 + r(\omega )} ),$

(3.11)

其中 $r(\omega)$是(2.5)式中的缓增函数,$M$是一依赖于$\lambda$ 的正常数,但是不依赖于$\omega$,$B$和$T_1$.

证 该结论直接由 (3.10)式和下式可得,对所有的 $T_1>0$,有

$\int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} \left\| {v ( {s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1({\Bbb R}^n) }^2 }{\rm d}s \\ \le\int_{0 }^t {{\rm e}^{\sigma ( {s - t} )} \left\| {v ( {s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1({\Bbb R}^n) }^2 }{\rm d}s. $

证毕.

引理3.3 假设条件(2.2)-(2.3)成立. 设 $\{B(\omega)\}_{\omega\in \Omega } \in {\cal D}$ 和 $v_0(\omega)\in B(\omega)$. 那么对每一个 $\omega \in \Omega $,存在$T=T(B,\omega)>0$,以致方程(2.1)的解$u(t,\omega,u_0({\omega}))$ 和$ v(t,\omega,v_0({\omega}))$,其中$u_0(\omega)=v_0(\omega)+z(\omega)$,满足对任意$t\geq T$ 和 $s\in [-t,0]$,有

$\left\| {u ( {t+s,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)}+ \left\| {v ( {t+s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} \\ \leq M{\rm e}^{-\sigma s}( { 1 + r^{p-1}(\omega )}) ,$

(3.12)

其中$r(\omega)$是(2.5)式中的缓增函数,$M$是一依赖于$\lambda$ 的正常数,但是不依赖于$\omega$ 和 $B$.

证 先在 (3.8)式两边同乘 ${\rm e}^{\frac{\sigma}{p-1} t}$,然后在 $(0,t+s)$上积分,其中 $s\in [-t,0]$,那么对任意 $t\geq 0,$ 有

$\left\| {v ( {t+s,\omega ,v_0 ( \omega )} )} \right\|^2_{H^1({\Bbb R}^n)} \le {\rm e}^{ - \frac{\sigma}{p-1} (t+s)} \left\| {v_0 (\omega )} \right\|^2_{H^1({\Bbb R}^n)} + \int_0^{t+s} {{\rm e}^{\frac{\sigma}{p-1} ( {\tau - (t+s)} )} } p_1 ( {\theta _\tau \omega } ){\rm d}\tau + \frac{pc}{\sigma }.$

用$\theta_{-t}\omega$替换$\omega$,由上式,(2.5)式以及 $r(\omega)$的缓增性,可得

$\left\| {v ( {t+s,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right\|^2_{H^1({\Bbb R}^n)} \\ \le {\rm e}^{ - \frac{\sigma}{p-1} (t+s)} \left\| {v_0 ( \theta_{-t}\omega )} \right\|^2_{H^1({\Bbb R}^n)} + \int_0^{t+s} {{\rm e}^{\frac{\sigma}{p-1} ( {\tau - (t+s)} )} } p_1 ( {\theta _{\tau-t} \omega } ){\rm d}\tau + \frac{pc}{\sigma } \\ = {\rm e}^{ - \frac{\sigma}{p-1} (t+s)} \left\| {v_0 ( \theta_{-t}\omega )} \right\|^2_{H^1({\Bbb R}^n)} + \int_{-t}^s {{\rm e}^{\frac{\sigma}{p-1} ( {\tau-s } )} } p_1 ( {\theta _{\tau} \omega } ){\rm d}\tau + \frac{pc}{\sigma } \\ \leq {\rm e}^{ - \frac{\sigma}{p-1} s} \bigg [{\rm e}^{ - \frac{\sigma}{p-1}t} \left\| {v_0 ( \theta_{-t}\omega )} \right\|^2_{H^1({\Bbb R}^n)} + \int_{-t}^0 {{\rm e}^{\frac{\sigma}{p-1}\tau} } p_1 ( {\theta _{\tau} \omega } ){\rm d}\tau + \frac{pc}{\sigma }\bigg] \\ \leq {\rm e}^{ - \frac{\sigma}{p-1} s} \bigg [{\rm e}^{ - \frac{\sigma}{p-1}t} \left\| {v_0 ( \theta_{-t}\omega )} \right\|^2_{H^1({\Bbb R}^n)} + \int_{-t}^0 {{\rm e}^{\frac{\sigma}{2(p-1)}\tau} }r(\omega){\rm d}\tau + \frac{pc}{\sigma } \bigg] \\ \leq {\rm e}^{ - \frac{\sigma}{p-1} s} \bigg[{\rm e}^{ - \frac{\sigma}{p-1}t} \left\| {v_0 ( \theta_{-t}\omega )} \right\|^2_{H^1({\Bbb R}^n)} + \frac{2(p-1)}{\sigma}r(\omega) + \frac{pc}{\sigma }\bigg].$

(3.13)

由假设,$\{B(\omega)\}_{\omega\in \Omega } \in {\cal D}$ 是缓增的. 因此,如果 $ v_0 ( {\theta _{ - t} \omega } ) \in B( {\theta _{ - t} \omega } ) $,那么存在 $T=T(B,\omega)>0$,以致对任意 $t\geq T$,

$\left\| {v ( {t+s,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right\|^2_{H^1({\Bbb R}^n)} \le c{\rm e}^{ - \frac{\sigma}{p-1} s}(1+r(\omega)).$

那么,

$\left\| {v ( {t+s,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right\|^{2(p-1)}_{H^1({\Bbb R}^n)} \le c{\rm e}^{ - {\sigma} s}(1+r^{p-1}(\omega)).$

(3.14)

注意到 $u( {t,\omega ,u_0 } ) = v( {t,\omega ,u_0 - z(\omega )} ) + z(\theta_t\omega )$. 故由 (3.14) 式和 $|y(\omega)|$ 的缓增性可得,对任意 $t\geq T$,

$\left\| {u( {t + s,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} \\ = \left\| {v( {t + s,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } ) - z( {\theta _{ - t} \omega } )} ) + z( {\omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} \\ \leq c( {\left\| {v( {t + s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| {z( { \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} } ) \\ \leq c{\rm e}^{ - {\sigma} s}(1+r^{p-1}(\omega)).$

故得证.

引理3.4 假设条件(2.2)-(2.3)成立. 设 $\{B(\omega)\}_{\omega\in \Omega } \in {\cal D}$ 和 $v_0(\omega)\in B(\omega)$. 那么方程(2.1)的解$u(t,\omega,u_0({\omega}))$ 和$ v(t,\omega,v_0({\omega}))$,其中$u_0(\omega)=v_0(\omega)+z(\omega)$,满足对任意$t\geq0$和每一个 $\omega \in \Omega $,有

$\left\| {v_t ( {t,\omega ,v_0 (\omega )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 \\ \le c( {\left\| {u( {t,\omega ,u_0 ( \omega )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| {v( {t,\omega ,v_0 (\omega )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left| {y(\theta_t\omega )} \right|^2 + 1} ).$

(3.15)

证 对方程 (2.6) 和 $v_t$ 在 $L^2({\Bbb R}^n)$上取内积,可得

$\left\| {v_t } \right\|^2 + \left\| {\nabla v_t } \right\|^2 = ( {\Delta v,v_t } ) - \lambda ( {v,v_t } ) + ( {f( {x,v + z(\theta_t\omega )} ),v_t } ) \\ \mbox{} + ( {g(x),v_t } ) + (1-\lambda) ( {\Delta z(\theta_t\omega ),v_t } ) \\ \le \frac{1}{4}\left\| {v_t } \right\|^2 + \frac{1}{4}\left\| {\nabla v_t } \right\|^2 + c( {\left\| v \right\|^2 + \left\| {\nabla v} \right\|^2 + \left\| {\nabla z(\theta_t\omega )} \right\|^2 + 1} ) \\ \mbox{}+ ( {f( {x,v + z(\theta_t\omega )} ),v_t } ).$

(3.16)

由(2.3) 式可得

$( {f( {x,v + z(\theta_t\omega )} ),v_t } ) \le \alpha _2 \int_{{\Bbb R}^n } {\left| u \right|^{p-1} } \left| {v_t } \right|{\rm d}x + \int_{{\Bbb R}^n } {\left| {\psi _2 } \right|} \left| {v_t } \right|{\rm d}x \\ \leq c\left\| u \right\|_{p }^{p-1} \left\| {v_t } \right\|_{p } + \left\| {\psi _2 } \right\|\left\| {v_t } \right\| \\ \le c\left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{p-1} \left\| {v_t } \right\|_{H^1 ( {{\Bbb R}^n } )} + \left\| {\psi _2 } \right\|\left\| {v_t } \right\| \\ \le \frac{1}{4}\left\| {v_t } \right\|^2 + \frac{1}{4}\left\| {\nabla v_t } \right\|^2 + c( {\left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + 1} ).$

(3.17)

由 (3.16)和 (3.17) 式可知

$\left\| {v_t } \right\|^2 + \left\| {\nabla v_t } \right\|^2 \le c( {\left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| v \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| {\nabla z(\theta_t\omega )} \right\|^2 + 1} )\\ \le c( {\left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| v \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + | { y(\theta_t\omega )} |^2 + 1} ).$

故得证.

引理3.5 假设条件(2.2)-(2.3)成立. 设 $\{B(\omega)\}_{\omega\in \Omega } \in {\cal D}$ 和 $v_0(\omega)\in B(\omega)$. 那么对任意$\varepsilon >0$ 和 $\omega\in \Omega $,存在$T=T(B,\omega,\varepsilon )\geq1$ 和 $K=K(\omega,\varepsilon )\geq1$ 以致对任意 $t\geq T$,方程 (2.6)的解$v$满足

$\int_{\left| x \right| \ge K} {\left| {v( {t ,\theta _{ - t } \omega ,v_0(\theta _{ - t } \omega) } )(x)} \right|} ^2 {\rm d}x \le \varepsilon . $

证 设 $\rho$为${\Bbb R}^+$上的光滑函数,并且满足对任意$s\in {\Bbb R}^+$,$0 \leq \rho(s) \leq 1$,特别的,当$0\leq s\leq1$时 $\rho(s)=0$,当$ s\leq 2$时$\rho(s)=1$. 显然$\rho'$在${\Bbb R}^+$上有界. 在方程(2.6)两边同乘$\rho ( {\frac{{\left| x \right|^2 }}{{k^2 }}} )v $,然后积分,可得

$\frac{1}{2}\frac{\rm d}{{\rm d}t}\int_{{\Bbb R}^n} {\rho \Big ( {\frac{{\left| x \right|^2 }} {{k^2 }}} \Big) (\left| {v } \right|^2 +\left|\nabla {v } \right|^2 )}{\rm d}y - \int_{{\Bbb R}^n} {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )\triangle v\cdot v } {\rm d}x + \lambda \int_{{\Bbb R}^n} {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)\left| {v } \right|^2 } {\rm d}x \\ = \int_{{\Bbb R}^n} {f ( {x,u } ) \rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)v } {\rm d}x + \int_{{\Bbb R}^n} {( {g ( {x} ) + (1-\lambda)\bigtriangleup z( {\theta _{2,t} \omega } )} )\rho \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)v } {\rm d}x.$

(3.18)

下面估计 (3.18) 式中的项. 首先有

$- \int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} \Delta v \cdot v{\rm d}x = \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} \left| {\nabla v} \right|^2 {\rm d}x + \int_{{\Bbb R}^n } {\rho ' \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \frac{{2x}}{{k^2 }}\nabla v \cdot v{\rm d}x \\ = \int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} \left| {\nabla v} \right|^2 {\rm d}x + \int_{k \le \left| x \right| \le \sqrt 2 k} {\rho ' \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \frac{{2x}}{{k^2 }}\nabla v \cdot v{\rm d}x. $

(3.19)

(3.19)式的右边第二项

$\left| {\int_{k \le \left| x \right| \le \sqrt 2 k} {\rho ' \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \frac{{2x}}{{k^2 }}\nabla v \cdot v{\rm d}x} \right| \leq \frac{{2\sqrt 2 }}{k}\int_{k \le \left| x \right| \le \sqrt 2 k} {\rho ' \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} \left| {\nabla v} \right| \cdot \left| v \right|{\rm d}x \\ \leq \frac{c}{k}\int_{k \le \left| x \right| \le \sqrt 2 k} {\left| {\nabla v} \right|} \cdot \left| v \right|{\rm d}x \\ \le \frac{c}{k}( {\left\| v \right\|^2 + \left\| {\nabla v} \right\|^2 } ).$

(3.20)

对非线性项,可得

$\int_{{\Bbb R}^n } {f( {x,u} )\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big )} v{\rm d}x = \int_{{\Bbb R}^n } {f( {x,u} )\rho\Big ( {\frac{{\left| x \right|^2 }} {{k^2 }}}\Big )} u{\rm d}x - \int_{{\Bbb R}^n } {f( {x,u} )\rho \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} z(\theta_t\omega ){\rm d}x.$

(3.21)

由(2.2),(3.21)式右边第一项,有下列估计

$\int_{{\Bbb R}^n } {f( {x,u} )\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big )} u{\rm d}x \le - \alpha _1 \int_{{\Bbb R}^n } {\left| u \right|^{p } \rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} {\rm d}x + \int_{{\Bbb R}^n } {\psi _1 \rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} {\rm d}x.$

(3.22)

由 (2.3),(3.21)式右边第二项,有下列估计

$\left| {\int_{{\Bbb R}^n } {f( {x,u} )\rho\Big ( {\frac{{\left| x \right|^2 }} {{k^2 }}}\Big )} z(\theta_t\omega ){\rm d}x} \right| \\ \le \alpha _2 \int_{{\Bbb R}^n } {\left| u \right|^{p-1} \rho\Big ( {\frac{{ \left| x \right|^2 }}{{k^2 }}} \Big)\left| {z( {\theta _t \omega } )} \right|} {\rm d}x + \int_{{\Bbb R}^n } {\left| {\psi _2 } \right|\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} \left| {z(\theta_t\omega )} \right|{\rm d}x \\ \le \frac{1}{2}\alpha _1 \int_{{\Bbb R}^n } {\left| u \right|^{p} \rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} {\rm d}x + c\int_{{\Bbb R}^n } {\left| {z(\theta_t\omega )} \right|^{p} \rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} {\rm d}x \\ \mbox{} + \frac{1}{2}\int_{{\Bbb R}^n } {\left| {\psi _2 } \right|^2 \rho \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} {\rm d}x + \frac{1}{2}\int_{{\Bbb R}^n } {\left| {z(\theta_t\omega )} \right|^2 \rho \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} {\rm d}x.$

(3.23)

故由(3.21)-(3.23) 式可知

$\int_{{\Bbb R}^n } {f( {x,u} )\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big )} v{\rm d}x \le - \frac{1}{2}\alpha _1 \int_{{\Bbb R}^n } {\left| u \right|^{p } \rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} {\rm d}x + \int_{{\Bbb R}^n } {\left| {\psi _1 } \right|\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} {\rm d}x \\ + \frac{1}{2}\int_{{\Bbb R}^n } {\left| {\psi _2 } \right|^2 \rho \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} {\rm d}x \\ + \int_{{\Bbb R}^n } {( {\left| {z(\theta_t\omega )} \right|^{p } + \left| {z(\theta_t\omega )} \right|^2 } ) \rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} {\rm d}x.$

(3.24)

另外,(3.18)式的右边最后一项满足

$ \int_{{\Bbb R}^n} {( {g +(1-\lambda) \bigtriangleup z( {\theta _{t} \omega } )} )\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )v } {\rm d}x \\ \le \frac{1}{2}\lambda \int_{{\Bbb R}^n}{\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )\left| {v } \right|^2 } {\rm d}x + c\int_{{\Bbb R}^n} {( {g^2 + \left| {\bigtriangleup z( {\theta _{t} \omega } )} \right|^2 } )\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} {\rm d}x.$

(3.25)

由 (3.18)-(3.20),(3.24) 和(3.25)式可得

$\frac{\rm d}{{\rm d}t}\int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big )} ( {\left| v \right|^2 + \left| {\nabla v} \right|^2 } ){\rm d}x + 2\int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \left| {\nabla v} \right|^2 {\rm d}x \\ \mbox{}+ \lambda \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big )} \left| v \right|^2 {\rm d}x + \alpha _1 \int_{{\Bbb R}^n } { \rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)\left| u \right|^{p}} {\rm d}x \\ \le \frac{c}{k}( {\left\| v \right\|^2 + \left\| {\nabla v} \right\|^2 } ) + c\int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} ( {\left| {\psi _1 } \right| + \left| {\psi _2 } \right|^2 + g^2 } ){\rm d}x \\ \mbox{}+ c\int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {\Delta z(\theta_t\omega )} \right|^2 + \left| {z(\theta_t\omega )} \right|^2 + \left| {z(\theta_t\omega )} \right|^p } ){\rm d}x.$

该式结合(2.8) 式,可得

$\frac{\rm d}{{\rm d}t}\int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big )} ( {\left| v \right|^2 + \left| {\nabla v} \right|^2 } ){\rm d}x + \sigma\int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \left| ({ v} \right|^2 + \left| {\nabla v} \right|^2) {\rm d}x \\ \le \frac{c}{k}( {\left\| v \right\|^2 + \left\| {\nabla v} \right\|^2 } ) +c \int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {\psi _1 } \right| + \left| {\psi _2 } \right|^2 + g^2 } ){\rm d}x \\ \mbox{}+ c\int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {\Delta z(\theta_t\omega )} \right|^2 + \left| {z(\theta_t\omega )} \right|^2 + \left| {z(\theta_t\omega )} \right|^p } ){\rm d}x.$

(3.26)

由引理3.1可知,存在 $T_1 = T_1(B,\omega ) > 0$ 以致对任意 $t \geq T_1$,有

$\left\| {v( {t,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )} \le c( {1 + r(\omega )} ). $

对上式两边同乘${\rm e}^{\sigma t}$,然后在 $(T_1,t)$上积分 ,可得,当 $t\geq T_1$时,有

$\int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {v( {t,\omega ,v_0 (\omega )} )} \right|^2 + \left| {\nabla v( {t,\omega ,v_0 (\omega )} )} \right|^2 } ){\rm d}x \\ \le {\rm e}^{\sigma ( {T_1 - t} )} \int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {v( {T_1 ,\omega ,v_0 (\omega )} )} \right|^2 + \left| {\nabla v( {T_1 ,\omega ,v_0 (\omega )} )} \right|^2 } ){\rm d}x \\ \mbox{}+ \frac{c}{k}\int_{T_1 }^t {{\rm e}^{\sigma( {s - t} )} } ( {\left\| {v( {s,\omega ,v_0 (\omega )} )} \right\|^2 + \left\| {\nabla v( {s,\omega ,v_0 (\omega )} )} \right\|^2 } ){\rm d}s \\ \mbox{} +c \int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} ( {\left| {\psi _1 } \right| + \left| {\psi _2 } \right|^2 + g^2 } ){\rm d}x{\rm d}s \\ \mbox{}+ c\int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } \int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} ( {\left| {\Delta z(\theta_t\omega )} \right|^2 + \left| {z( {\theta _t \omega } )} \right|^2 + \left| {z(\theta_t\omega )} \right|^p } ){\rm d}x{\rm d}s.$

(3.27)

在(3.27)式中用$\theta_{-t}\omega$代替$\omega$ 可得,当 $t\geq T_1$时,有

$\int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {v( {t,\theta _{ - t} \omega ,v_0 (\omega )} )} \right|^2 + \left| {\nabla v( {t,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right|^2 } ){\rm d}x \\ \le {\rm e}^{\sigma ( {T_1 - t} )} \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} ( {\left| {v( {T_1 ,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right|^2 + \left| {\nabla v( {T_1 ,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right|^2 } ){\rm d}x \\ \mbox{}+ \frac{c}{k}\int_{T_1 }^t {{\rm e}^{\sigma( {s - t} )} } ( {\left\| {v( {s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|^2 + \left\| {\nabla v( {s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|^2 } ){\rm d}s \\ \mbox{}+c \int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} ( {\left| {\psi _1 } \right| + \left| {\psi _2 } \right|^2 + g^2 } ){\rm d}x{\rm d}s \\ \mbox{}+ c\int_{T_1 }^t {{\rm e}^{\sigma( {s - t} )} } \int_{{\Bbb R}^n } {\rho \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {\Delta z( {\theta _{s - t} \omega } )} \right|^2 + \left| {z( {\theta _{s - t} \omega } )} \right|^2 + \left| {z( {\theta _{s - t} \omega } )} \right|^p } ){\rm d}x{\rm d}s.$

(3.28)

下面,估计 (3.28)式中的项. 首先在 (3.9)式中用$T_1$代替 $t$,然后用$\theta_{-t}\omega$代替$\omega$,(3.28)式右边第一项有如下估计

${\rm e}^{\sigma ( {T_1 - t} )} \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {v( {T_1 ,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right|^2 + \left| {\nabla v( {T_1 ,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right|^2 } ){\rm d}x \\ \le {\rm e}^{\sigma ( {T_1 - t} )} ( {{\rm e}^{ - \sigma T_1 } \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 + \int_0^{T_1 } {{\rm e}^{\lambda ( {s - T_1 } )} p_1 ( {\theta _{s - t} \omega } ){\rm d}s + c} } ) \\ \le {\rm e}^{ - \sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 + c{\rm e}^{\sigma ( {T_1 - t} )} + \int_{ - t}^{T_1 - t} {{\rm e}^{\sigma \tau } p_1 ( {\theta _\tau \omega } ){\rm d}\tau } \\ \le {\rm e}^{ - \lambda t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 + c{\rm e}^{\sigma ( {T_1 - t} )} + c\int_{ - t}^{T_1 - t} {{\rm e}^{\frac{1}{2}\sigma \tau } r(\omega ){\rm d}\tau } \\ \le {\rm e}^{ - \sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 + c{\rm e}^{\sigma ( {T_1 - t} )} + \frac{2}{\sigma} cr(\omega ){\rm e}^{\frac{1}{2}\sigma ( {T_1 - t} )},$

(3.29)

其中,我们用到了 (3.7)式. 由 (3.29)式可知,给定 $\varepsilon >0$,存在 $T_2 = (B,\omega,\varepsilon ) > T_1$,以致对任意 $t \geq T_2$,

${\rm e}^{\sigma ( {T_1 - t} )} \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }} {{k^2 }}}\Big )} ( {\left| {v( {T_1 ,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right|^2 + \left| {\nabla v( {T_1 ,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right|^2 } ){\rm d}x \le \varepsilon .$

(3.30)

由引理 3.2,存在 $T_3 = T_3(B,\omega) > T_1$,以致 (3.28)式右边第二项满足当 $t \geq T_3$时,

$\frac{c}{k}\int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } ( {\left\| {v ( {s,\theta_{-t}\omega ,v_0 ( \theta_{-t}\omega )} )} \right\|^2 + \left\| {\nabla v( {s,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right\|^2 } ){\rm d}s \le \frac{c}{k}( {1 + r(\omega )} ). $

因此,存在 $R_1 = R_1(\omega,\varepsilon ) > 0$,对致对任意 $t \geq T_3$和 $k\geq R_1$,有

$\frac{c}{k}\int_{T_1 }^t {{\rm e}^{\sigma( {s - t} )} } ( {\left\| {v( {s,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right\|^2 + \left\| {\nabla v( {s,\theta_{-t}\omega ,v_0 ( \theta_{-t}\omega )} )} \right\|^2 } ){\rm d}s \le \varepsilon .$

(3.31)

注意到 $\psi_1 \in L^1({\Bbb R}^n)$ 和$\psi_2,g\in L^2({\Bbb R}^n)$. 因此,存在 $R_2 = R_2(\varepsilon )$ ,使得当$k\geq R_2$时,有

$c\int_{\left| x \right| \ge k} {( {\left| {\psi _1 } \right| + \left| {\psi _2 } \right|^2 + g^2 } )} {\rm d}x \le \sigma \varepsilon . $

(3.28)式右边第三项有如下估计

$ c\int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} ( {\left| {\psi _1 } \right| + \left| {\psi _2 } \right|^2 + g^2 } ){\rm d}x{\rm d}s \\ \leq c \int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } \int_{\left| x \right| \ge k} {( {\left| {\psi _1 } \right| + \left| {\psi _2 } \right|^2 + g^2 } )} {\rm d}x{\rm d}s \\ \le \sigma\varepsilon \int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} }{\rm d}s \le \varepsilon .$

(3.32)

由于 $(I-\Delta)^{-1}h \in H^2({\Bbb R}^n)\cap W^{2,p}({\Bbb R}^n)$,故存在 $R_3 =R_3(\omega,\varepsilon ) > 0$,当 $k\geq R_3$时,有

$\int_{\left| x \right| \ge k} {( {\left| {( {I - \Delta } )^{ - 1} h} \right|^2 + \left| {\Delta ( {I - \Delta } )^{ - 1} h} \right|^2 + \left| {( {I - \Delta } )^{ - 1} h} \right|^p } )} {\rm d}x \le \frac{{\sigma\varepsilon }}{{2c r(\omega )}}.$

(3.33)

注意到$z(\theta_t\omega ) = ( {I - \Delta } )^{ - 1} hy(\theta_t\omega )$. 由(3.33)式,(2.5)式和$r(\omega)$的缓增性,(3.28)式右边第四项满足

$c\int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {\Delta z ( {\theta _{s - t} \omega } )} \right|^2 + \left| {z( {\theta _{s - t} \omega } )} \right|^2 + \left| {z( {\theta _{s - t} \omega } )} \right|^p } ){\rm d}x{\rm d}s \\ \le c\int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } \int_{\left| x \right| \ge k} {( {\left| {\Delta z( {\theta _{s - t} \omega } )} \right|^2 + \left| {z( {\theta _{s - t} \omega } )} \right|^2 + \left| {z( {\theta _{s - t} \omega } )} \right|^p } )} {\rm d}x{\rm d}s \\ =c\int_{T_1 }^t {{\rm e}^{\sigma( {s - t} )} } \int_{\left| x \right| \ge k} {( {\left| {\Delta ( {I - \Delta } )^{ - 1} hy( {\theta _{s - t} \omega } )} \right|^2 + \left| {( {I - \Delta } )^{ - 1} hy( {\theta _{s - t} \omega } )} \right|^2 } )} {\rm d}x{\rm d}s \\ \mbox{}+ c\int_{T_1 }^t {{\rm e}^{\sigma( {s - t} )} } \int_{\left| x \right| \ge k} {\left| {( {I - \Delta } )^{ - 1} hy( {\theta _{s - t} \omega } )} \right|^p } {\rm d}x{\rm d}s \\ \le \frac{{\sigma\varepsilon }}{{ 2r(\omega )}}\int_{T_1 }^t {{\rm e}^{\sigma ( {s - t} )} } ( {\left| {y( {\theta _{s - t} \omega } )} \right|^2 + \left| {y( {\theta _{s - t} \omega } )} \right|^p } ){\rm d}s \\ \leq \frac{{\sigma\varepsilon }}{{2 r(\omega )}}\int_{T_1 }^t {{\rm e}^{\sigma( {s - t} )} } r( {\theta _{s - t} \omega } ){\rm d}s \le \frac{{\sigma\varepsilon }}{{2 r(\omega )}}\int_{T_1 - t}^0 {{\rm e}^{\sigma \tau } } r( {\theta _\tau \omega } ){\rm d}\tau \\ \le \frac{{\sigma\varepsilon }}{{2 r(\omega )}}\int_{T_1 - t}^0 {{\rm e}^{\frac{1}{2}\sigma \tau } } r(\omega ){\rm d}\tau \le \varepsilon .$

(3.34)

设 $T_4 =T_4(B,\omega,\varepsilon ) = \max\{T_1,T_2,T_3 \}$ 和 $ R_4= R_4(\omega,\varepsilon ) = \max\{R_1,R_2,R_3\}$. 那么由 (3.28),(3.30),(3.31),(3.32)和 (3.34)式可知,当 $t \geq T_4$ 和 $k \geq R_4$时,有

$\int_{{\Bbb R}^n } {\rho\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {v( {t,\omega ,v_0 (\omega )} )} \right|^2 + \left| {\nabla v( {t,\omega ,v_0 (\omega )} )} \right|^2 } ){\rm d}x \le 4\varepsilon . $

由上式可知,当 $t \geq T_4$ 和 $k \geq R_4$时,

$ \int_{\left| x \right| \ge \sqrt 2 k} {( {\left| {v( {t,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right|^2 + \left| {\nabla v( {t,\theta_{-t}\omega ,v_0 ( \theta_{-t}\omega )} )} \right|^2 } )} {\rm d}x \\ \leq \int_{{\Bbb R}^n } {\rho \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} ( {\left| {v( {t,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right|^2 + \left| {\nabla v( {t,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right|^2 } ){\rm d}x \le 4\varepsilon . $

故得证.

现在估计方程(2.6)的解在有界区域的的行为. 为此,记$\psi( s ) = 1-\rho( s )$,其中$\rho( s )$是引理3.5 中给定的函数$\rho( s )$. 设$k\geq1$及

$\tilde v = \psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )v.$

(3.35)

注意到$\tilde v\in H^1_0(Q_{2k})$,其中$Q_{2k}=\{x\in {\Bbb R}^n:|x|＜2k\}$. 在方程(2.6)两边同乘$ \psi ( {\frac{{\left| x \right|^2 }}{{k^2 }}} ) $,可得

$\tilde v_t - \Delta \tilde v_t - \Delta \tilde v + \lambda \tilde v = \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )f( {x,u} ) + \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)g(x) \\ \mbox{} + (1-\lambda) \psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big )\Delta z(\theta_t\omega ) -\Big ( {v_t \Delta \psi \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big ) + 2\nabla v_t \nabla \psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)}\Big ) \\ \mbox{} - \Big( {v\Delta \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big) + 2\nabla v\nabla \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} \Big).$

(3.36)

考虑下列特征值问题

$\left\{ \begin{array}{ll} - \Delta \tilde v = \lambda \tilde v,& \mbox{in} Q_{2k},\\ \tilde v = 0,&\mbox{on}tial Q_{2k}. \end{array} \right. $

则其对应的特征值和对应的特征函数为 $\left\{ {\lambda _j } \right\}_{j = 1}^\infty$和 $\left\{ {e_j } \right\}_{j = 1}^\infty$,其中 $\left\{ {e_j } \right\}_{j = 1}^\infty$为 $L^2 ( {Q_{2k} } )$中的一组正交基. 给定$n$,设$X_n = \mbox{span}\left\{ {e_1 ,e_2 ,\cdots e_n } \right\} $及$P_n :L^2 ( {Q_{2k} } ) \to X_n$是投影算子.

引理3.6 假设条件(2.2)-(2.3)成立. 设 $\{B(\omega)\}_{\omega\in \Omega } \in {\cal D}$ 和 $v_0(\omega)\in B(\omega)$. 那么对任意$\varepsilon >0$ 和 $\omega\in \Omega $,存在$T=T(B,\omega,\varepsilon )\geq1$,$K=K(\omega,\varepsilon )\geq1$ 和$N=N(\omega,\varepsilon )>0$以致方程 (2.6)的解$v(t,\omega,v_0({\omega}))$当 $t\geq T$,$k\geq K$ 和 $n\geq N$时,有

$\int_{\left| x \right| \ge K} {\left| {v( {t ,\theta _{ - t } \omega ,v_0(\theta _{ - t } \omega) } )(x)} \right|} ^2 {\rm d}x \le \varepsilon . $

证 设 $\tilde v_{n,1} = P_n \tilde v $ 和 $\tilde v_{n,2} =( I-P_n) \tilde v $. 对$\tilde v_{n,2}$在 $L^2 ( {Q_{2k} } )$上取内积,可得

$ \frac{1}{2}\frac{\rm d}{{\rm d}t}( {\left\| {\tilde v_{n,2} } \right\|^2 + \left\| {\nabla \tilde v_{n,2} } \right\|^2 } ) + \left\| {\nabla \tilde v_{n,2} } \right\|^2 + \lambda \left\| {\tilde v_{n,2} } \right\|^2 \\ = \Big( {\psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)f ( {x,u} ),\tilde v_{n,2} } \Big) +\Big ( {\psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)g(x),\tilde v_{n,2} } \Big) \\ \mbox{} + \Big( {(1-\lambda) \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big )\Delta z(\theta_t\omega ),\tilde v_{n,2} }\Big ) \\ \mbox{} - \Big( {v_t \Delta \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big ) + 2\nabla v_t \nabla \psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big),\tilde v_{n,2} } \Big) \\ \mbox{} - \Big( {v\Delta \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big ) + 2\nabla v\nabla \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big ),\tilde v_{n,2} } \Big).$

(3.37)

注意当$s \notin ( 1 ,2 ) $时,$\psi' ( s ) = 0$. 故有

$\left| {\nabla \psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \right| = \left| {\psi' \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )} \right|\frac{{2\left| x \right|}}{{k^2 }} \le \left| {\psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \right|\frac{{2\sqrt 2 k}}{{k^2 }} \le \frac{c}{k}$

(3.38)

和

$\left| {\Delta \psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \right| = \left| {\psi " \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)\frac{{4\left| x \right|^2 }}{{k^4 }} + \psi ' \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )\frac{2}{{k^2 }}} \right| \\ \le \left| {\psi " \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)} \right|\frac{{8k^2 }}{{k^4 }} + \left| {\psi ' \Big( {\frac{{\left| x \right|^2 }} {{k^2 }}}\Big )} \right|\frac{2}{{k^2 }} \le \frac{c}{{k^2 }}.$

(3.39)

由 (3.38)-(3.39)式,(3.37)式的右边项有如下估计

$\Big( {\psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )g(x),\tilde v_{n,2} }\Big ) \le \frac{1}{4}\left\| {\nabla \tilde v_{n,2} } \right\|^2 + c\lambda _{n + 1}^{ - 1} \left\| g \right\|^2 ,$

(3.40)

$\Big ( {(1-\lambda) \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )\Delta z(\theta_t\omega ),\tilde v_{n,2} } \Big) \le \frac{1}{4}\left\| {\nabla \tilde v_{n,2} } \right\|^2 + c\lambda _{n + 1}^{ - 1} \left\| {\Delta z(\theta_t\omega )} \right\|^2 \\ = \frac{1}{4}\left\| {\nabla \tilde v_{n,2} } \right\|^2 + c\lambda _{n + 1}^{ - 1} \left\| {\Delta ( {I - \Delta } )^{ - 1} hy(\theta_t\omega )} \right\|^2 \\ \le \frac{1}{4}\left\| {\nabla \tilde v_{n,2} } \right\|^2 + c\lambda _{n + 1}^{ - 1} \left| {y(\theta_t\omega )} \right|^2 ,$

(3.41)

$\Big( {v_t \Delta \psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big ) + 2\nabla v_t \nabla \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big),\tilde v_2 } \Big) \le \frac{\lambda }{4}\left\| {\tilde v_2 } \right\|^2 + \frac{c}{{k^4 }}\left\| {v_t } \right\|^2 + \frac{c}{{k^2 }}\left\| {\nabla v_t } \right\|^2$

(3.42)

和

$\Big( {v\Delta \psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big ) + 2\nabla v\nabla \psi \Big( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big),\tilde v_2 } \Big ) \le \frac{\lambda }{4}\left\| {\tilde v_2 } \right\|^2 + \frac{c}{{k^4 }}\left\| v \right\|^2 + \frac{c}{{k^2 }}\left\| {\nabla v} \right\|^2.$

(3.43)

设 $\theta = \frac{{n ( {p - 2} )}}{{2p}}$. 由于若 $n\geq 3$,则$2\leq p＜\frac{2n}{n-2}$,可知 $0\leq\theta＜1$. 故由 (2.3)式和内插不等式,可知 (3.37)式的右边第一项满足

$\Big ( {\psi\Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}} \Big)f( {x,u} ),\tilde v_{n,2} } \Big) \le \alpha _2 \int_{{\Bbb R}^n } {\psi \Big ( {\frac{{\left| x \right|^2 }}{{k^2 }}}\Big )\left| u \right|^{p-1} \left| {\tilde v_{n,2} } \right|} {\rm d}x + \int_{{\Bbb R}^n } {\psi \Big( {\frac{{\left| x \right|^2 }} {{k^2 }}} \Big)\psi _2 (x)\left| {\tilde v_{n,2} } \right|} {\rm d}x \\ \leq c\left\| u \right\|_{p}^{p-1}\left\| {\tilde v_{n,2} } \right\|_{p} + \left\| {\psi _2 } \right\|\left\| {\tilde v_{n,2} } \right\| \\ \leq c\left\| u \right\|_{p}^{p-1} \left\| {\nabla \tilde v_{n,2} } \right\|^\theta \left\| {\tilde v_{n,2} } \right\|^{1 - \theta } + \lambda _{n + 1}^{ - \frac{1}{2}} \left\| {\psi _2} \right\|\left\| {\nabla \tilde v_{n,2} } \right\| \\ \leq c\lambda _{n + 1}^{\frac{{\theta - 1}}{2}} \left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{p-1} \left\| {\nabla \tilde v_{n,2} } \right\| + \lambda _{n + 1}^{ - \frac{1}{2}} \left\| {\psi _2 } \right\|\left\| {\nabla \tilde v_{n,2} } \right\| \\ \leq \frac{1}{4}\left\| {\nabla \tilde v_{n,2} } \right\|^2 + c\lambda _{n + 1}^{\theta - 1} \left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} +c \lambda _{n + 1}^{ - 1} \left\| {\psi _2 } \right\|^2.$

(3.44)

由(3.40)-(3.44)式和 (3.37)式,有

$\frac{\rm d}{{\rm d}t}( {\left\| {\tilde v_{n,2}} \right\|^2 + \left\| {\nabla \tilde v_{n,2} } \right\|^2 } ) + \frac{1}{2}\left\| {\nabla \tilde v_{n,2} } \right\|^2 + \lambda \left\| {\tilde v_{n,2} } \right\|^2 \\ \le c\lambda _{n + 1}^{\theta - 1} \left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} +c \lambda _{n + 1}^{ - 1} \left\| {\psi _2} \right\|^2 + c\lambda _{n + 1}^{ - 1} \left\| g \right\|^2 \\ \mbox{}+ c\lambda _{n + 1}^{ - 1} \left| {y( {\theta _t \omega } )} \right|^2 + \frac{c}{{k^4 }}\left\| {v_t } \right\|^2 + \frac{c}{{k^2 }}\left\| {\nabla v_t } \right\|^2 + \frac{c}{{k^4 }} \left\| v \right\|^2 + \frac{c}{{k^2 }}\left\| {\nabla v} \right\|^2.$

(3.45)

注意$\psi_2,g\in L^2({\Bbb R}^n)$和 $\theta＜1$. 由引理3.4,(2.8)和 (3.45)式,对任意 $\varepsilon >0$,存在 $N = N(\varepsilon )$ 和 $K= K(\varepsilon )$,以致对任意 $n \geq N$ 和$k\geq K$,有

$\frac{\rm d}{{\rm d}t}( {\left\| {\tilde v_{n,2} } \right\|^2 + \left\| {\nabla \tilde v_{n,2} } \right\|^2 } ) +2\sigma (\left\| {\nabla \tilde v_{n,2} } \right\|^2 + \left\| {\tilde v_{n,2} } \right\|^2) \\ \le c\varepsilon ( {1 + \left| {y(\theta_t\omega )} \right|^2 + \left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| v_t \right\|_{H^1({\Bbb R}^n)}^{2} + \left\| v \right\|_{H^1({\Bbb R}^n)}^{2} } ) \\ \le c\varepsilon ( {1 + \left| {y(\theta_t\omega )} \right|^2 + \left\| u \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| v \right\|_{H^1({\Bbb R}^n)}^{2(p-1)} } ).$

(3.46)

对上式两边同乘 ${\rm e}^{2\sigma t}$,然后$(0,t)$上积分,可得,对任意$t\geq 0,$ 有

$\left\| {\tilde v_{n,2} ( {t,\omega ,v_0 (\omega )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 \leq {\rm e}^{ - 2\sigma t} \left\| v_0 (\omega ) \right\|_{H^1 ( {\Bbb R}^n )} + c\varepsilon \int_0^t {\rm e}^{ - 2\sigma ( {t - s} )} ( 1 + \left| y( {\theta _s \omega } ) \right|^2 \\ + \left\| u( s,\omega ,u_0 (\omega ) ) \right\|_{H^1 ( {\Bbb R}^n )}^{2(p-1)} + \left\| v( s,\omega ,v_0 (\omega ) ) \right\|_{H^1 ( {\Bbb R}^n )}^{2(p-1)} ){\rm d}s . $

(3.47)

在(3.47)式中用$\theta_{-t}\omega$代替 $\omega$,由引理3.3可得,当$t\geq 0$时,有

$ \left\| {\tilde v_{n,2} ( {t,\theta _{ - t} \omega ,v_0 (\omega )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 \\ \leq {\rm e}^{ - 2\sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )} + c\varepsilon \int_0^t {{\rm e}^{ - 2\sigma ( {t - s} )} ( {1 + \left| {y( {\theta _{s - t} \omega } )} \right|^2 } )}{\rm d}s \\ \mbox{}+ c\varepsilon \int_0^t {{\rm e}^{ - 2\sigma ( {t - s} )} ( {\left\| {u ( {s,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| {v( {s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} } )}{\rm d}s \\ \le {\rm e}^{ - 2\sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )} + c\varepsilon \int_{ - t}^0 {{\rm e}^{2\sigma s} ( {1 + \left| {y( {\theta _s \omega } )} \right|^2 } )}{\rm d}s \\ \mbox{}+ c\varepsilon \int_{ - t}^0 {{\rm e}^{2\sigma s} ( {\left\| {u ( {t + s,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} + \left\| {v( {t + s,\theta _{ - t} \omega ,v_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^{2(p-1)} } )}{\rm d}s \\ \le {\rm e}^{ - 2\sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )} + c\varepsilon \int_{ - t}^0 {{\rm e}^{2\sigma s} ( {1 + \left| {y( {\theta _s \omega } )} \right|^2 } )}{\rm d}s + c\varepsilon \int_{ - t}^0 {{\rm e}^{\sigma s} }{\rm d}s( {1 + r^{p-1} (\omega )} ) \\ \le {\rm e}^{ - 2\sigma t} \left\| {v_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1 ( {{\Bbb R}^n } )} + c\varepsilon ( {1 + r^{p-1} (\omega )} ).$

(3.48)

由于 $v_ 0(\theta_{-t}\omega) \in B (\theta_{-t}\omega)$,(3.48)式的右边第一项当$t\rightarrow \infty$时渐近于零. 那么存在 $T = T( B ,\omega,\varepsilon ) > 0$,以致对所有 $t\geq T_1$,第一项小于 $\varepsilon $. 故由(3.48)式,当$n \geq N ,k \geq K$ 和 $t\geq T$时,有

$\left\| {\tilde v_{n,2} ( {t,\theta_{-t}\omega ,v_0 (\theta_{-t} \omega )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 \le c\varepsilon ( {1 + r^{p-1} (\omega )} ).$

故得证.

下面,证明主要结果. 注意到 $u( {t,\omega ,u_0 } ) = v( {t,\omega ,u_0 - z(\omega )} ) + z(\theta_t\omega ). $ 故由 (3.10)式可知,对所有$t\geq0$,有

$\left\| {u ( {t,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1({\Bbb R}^n) }^2 \\ = \left\| {v ( {t,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } ) - z( {\theta _{ - t} \omega } )} ) + z(\omega )} \right\|_{H^1({\Bbb R}^n) }^2 \\ \le 2\left\| {v ( {t,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } ) - z( {\theta _{ - t} \omega } )} )} \right\|_{H^1({\Bbb R}^n) }^2 + 2\left\| {z(\omega )} \right\|_{H^1({\Bbb R}^n) }^2 \\ \le 2{\rm e}^{ - \sigma t} \left\| {u_0 ( {\theta _{ - t} \omega } ) - z( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 + cr( \omega ) + c + 2\left\| {z(\omega )} \right\|_{H^1({\Bbb R}^n) }^2 \\ \le 4{\rm e}^{ - \sigma t} \left\| {u_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 + 4{\rm e}^{ - \sigma t} \left\| {z( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2\\ \mbox{}+ cr(\omega ) + c + 2\left\| {z(\omega )} \right\|_{H^1({\Bbb R}^n) }^2.$

(4.1)

因此,如果 $ u_0 ( {\theta _{ - t} \omega } ) \in B( {\theta _{ - t} \omega } )\in {\cal D} $,则存在 $T=T(B,\omega)>0$,当 $t\geq T$时,有

$4{\rm e}^{ - \sigma t} ( {\left\| {u_0 ( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2 + \left\| {z( {\theta _{ - t} \omega } )} \right\|_{H^1({\Bbb R}^n) }^2} ) \le cr(\omega ) + c. $

上式结合 (4.1) 式可得,当 $t\geq T$时,有

$\left\| {u ( {t,\theta _{ - t} \omega ,u_0 ( {\theta _{ - t} \omega } )} )} \right\|_{H^1({\Bbb R}^n) }^2 \le 2( {cr(\omega ) + c + \left\| {z(\omega )} \right\|_{H^1({\Bbb R}^n) }^2 } ).$

(4.2)

给定 $\omega\in \Omega $,记

$K(\omega ) = \left\{ {u \in H^1({\Bbb R}^n): \left\| {u } \right\|_{H^1({\Bbb R}^n) }^2 \le 2( {cr(\omega ) + c + \left\| {z(\omega )} \right\|_{H^1({\Bbb R}^n) }^2 } )} \right\}.$

(4.3)

那么 $\{K(\omega )\}_{\omega\in \Omega } \in {\cal D}$. 更进一步,由(4.3) 式可得 $\{K(\omega )\}_{\omega\in \Omega }$ 是$\phi$在${\cal D}$中的一吸收集.

下面证明$\phi$在$H^1({\Bbb R}^n)$中是拉回渐近紧的.

引理4.1 假设条件 (2.2)-(2.3) 成立. 那么随机动力系统$\phi$是在$H^1({\Bbb R}^n)$中$\mathcal D$ -拉回拉回渐近紧的,即当每一个$\omega\in \Omega $,系列 $\left\{ {\phi ( {t_m ,\theta _{ - t_m } \omega ,u_{0,m} ( {\theta _{ - t_m } \omega } )} )} \right\}$ 在$H^1({\Bbb R}^n)$中存在收敛子列,其中$t_m \rightarrow \infty$,$ B = \left\{ {B(\omega )} \right\}_{\omega \in \Omega } \in \mathcal D $ 和 $u_{0,m} ( {\theta _{ - t_m } \omega } ) \in B( {\theta _{ - t_m } \omega } ).$

证 由于 $t_m\rightarrow 0$,由引理 3.1可知,对每一个$\omega\in \Omega $,存在 $M_1=M_1(B,\omega)$,当 $m\geq M_1$时,有

$\left\| {v( {t_n,\theta _{ - t_m} \omega ,v_0 ( {\theta _{ - t_m} \omega } )} )} \right\|_{H^1 ( {{\Bbb R}^n } )}^2 \le c(1+r(\omega)).$

(4.4)

给定 $\varepsilon >0$,由引理 3.5,存在 $M_2=M_2(B,\omega,\varepsilon )$ 和 $k_0=k_0(\omega,\varepsilon )$,当 $m\geq M_2$时,有

$\int_{{\Bbb R}^n \backslash Q_{k_0 } } {( {\left| {v( {t_n ,\theta _{ - t_n } \omega ,v_0 ( {\theta _{ - t_m } \omega } )} )} \right| + \left| {\nabla v( {t_m ,\theta _{ - t_n } \omega ,v_0 ( {\theta _{ - t_m } \omega } )} )} \right|} )} {\rm d}x \le \varepsilon .$

(4.5)

设 $\tilde v$和(3.35)式中定义的一样. 那么由引理3.6可知,存在 $k_1=k_1(\omega,\varepsilon )\geq k_0$,$M_3=M_3(B,\omega,\varepsilon )\geq \max\{M_1,M_2\}$ 和 $N=N(\omega,\varepsilon )$,当$M\geq M_3$时,有

$\left\| {( {I - P_N } )\tilde v( {t_m ,\theta _{ - t_m } \omega ,v_0 ( {\theta _{ - t_m} \omega } )} )} \right\|_{H^1(Q_{2k_1})} \le \varepsilon .$

(4.6)

由(4.4)式可知 $\{P_N \tilde v( {t_m ,\theta _{ - t_m } \omega ,v_0 ( {\theta _{ - t_m } \omega } )} )\} $在有限维空间$P_N H^1 ( {Q_{2k_1 } } )$是有界的,故 \linebreak $ P_N H^1 ( {Q_{2k_1 } } )$是列紧的. 结合 (4.6)式可知 $\{\tilde v( {t_m ,\theta _{ - t_m } \omega ,v_0 ( {\theta _{ - t_m } \omega } )} )\} $ 在 $H^1 ( {Q_{2k_1 } } )$中是列紧的. 由于对$\left| x \right| \le k_1$有 $\tilde v( {t_m ,\theta _{ - t_m} \omega ,v_0 ( {\theta _{ - t_m } \omega } )} ) = v( {t_m ,\theta _{ - t_m } \omega ,v_0 ( {\theta _{ - t_m } \omega } )} ),$故

$\{ v( {t_m ,\theta _{ - t_m } \omega ,v_0 ( {\theta _{ - t_m } \omega } )} )\} \mbox{在$H^1 ( {Q_{k_1 } } )$中准紧.} $

上式结合 (4.5)式可知 $\{ v( {t_m ,\theta _{ - t_m } \omega ,v_0 ( {\theta _{ - t_m} \omega } )} )\}$ 在 $H^1 ( {\Bbb R}^n )$ 中是列紧的. 由 (2.7)式可知 $\phi$ 是在$H^1({\Bbb R}^n)$中 $\mathcal D$ -拉回渐近紧的. 故得证.

定理4.1 假设条件 (2.2)-(2.3) 成立,则 $\phi$在空间$H^1({\Bbb R}^n)$中存在唯一 ${\mathcal D}$ -随机吸引子 $\mathcal A$.

证 由(4.3)式可知 $\phi$有一闭吸收集. 引理4.1证明了 $\phi$ 是在$H^1({\Bbb R}^n)$中 $\mathcal D$ -拉回渐近紧的. 因此由文献[15,命题2.7]可得$\phi$ 在空间$H^1({\Bbb R}^n)$中存在唯一 ${\mathcal D}$ -随机吸引子 $\mathcal A$.