我们考虑3维的Benjamin-Bona-Mahony (BBM)方程

(1.1) $\left\{\begin{array}{ll} u_{t}-\triangle u_{t}-\nu\triangle u+\nabla\cdot\overrightarrow{F}(u)=g(x, t), &(x, t)\in\Omega\times(0, \infty), \\ u(x, t)\mid_{\partial\Omega}=0,&t\in(0, \infty), \\ u(x, 0)=u_{0}(x), &x\in\Omega, \end{array} \right.$

其中$\Omega\subset{\Bbb R}^{3}$是有足够光滑边界$\partial\Omega$的有界域, $u=u(x, t)=(u_{1}(x, t), u_{2}(x, t), u_{3}(x, t))$表示流速向量, $\nu>0$是动力粘度, $\overrightarrow{F}$是非线性向量函数, $g$是外力项.

BBM方程作为一种传播长波的模型,近几年来被一些作者广泛研究.文献[2]通过对无界域上BBM方程渐近性的研究得到了3维的BBM方程全局吸引子的存在性.文献[3]解决了BBM方程全局吸引子的正则性问题,并证明了外力项光滑时全局吸引子也是光滑的.由于BBM方程生成的动力系统在$H_{per}^{2}$不是紧的,故文献[4]首先得到了该方程存在一个弱的全局吸引子,接着通过能量估计证明了$H_{per}^{2}$中的弱全局吸引子在弱拓扑意义下几乎为强全局吸引子,从而证明了$H_{per}^{2}$中存在有限维的全局吸引子.文献[7]证明了广义BBM方程生成的动力系统在周期边界条件下全局吸引子的Gevrey正则性.文献[9]给出了BBM方程在$H^{1}(\Omega)$中拉回吸引子的存在性.文献[12]中作者得到了BBM方程在${\Bbb R}^{3}$中解的渐近性,并证明了该方程在$H^{1}({\Bbb R}^{3})$中吸引子的存在性,进一步得到了该吸引子在$H^{2}({\Bbb R}^{3})$中是有界的.文献[14]用弱收敛的思想在$H_{0}^{2}(\Omega)$中建立了过程${U(t, \tau)}$的渐近紧性,从而得到了BBM方程在$H^{2}(\Omega)$中拉回吸引子的存在性.文献[8]运用算子分解技巧和紧性平移定理分别证明了3维BBM方程(1.1)的全局吸引子和一致吸引子的存在性.文献[1]研究具有周期边界条件的多维广义BBM方程,获过得了具有有限分形维数全局吸引子的存在性和其相应半群指数吸引子的存在性.

我们在文献[8]的研究基础上,应用文献[6]中提出的加强的平坦性条件证明了自治BBM方程$(1.1)$ (即$g(x, t)$$=g(x)$)指数吸引子的存在性;进一步借助文献[5]中的方法和技巧获得了非自治系统指数吸引子的存在性.

引入与外力项$g(x, t)$有关的空间$L^2_{\rm loc}({\Bbb R};L^{k}(\Omega))$, $r, k\geq1$ :

$L^r_{b}({\Bbb R};L^k(\Omega)):=\bigg\{g\in{L^{r}_{\rm loc}({\Bbb R};L^{k}(\Omega))}:\sup\limits_{t\in{{\Bbb R}}}\int^{t+1}_{t}\bigg(\int_{\Omega}|g(x, s)|^{k}{\rm d}x\bigg)^{\frac{r}{k}}{\rm d}s<\infty\bigg\}, $

其中$ \|f\|^{2}_{L^{2}_{b}({\Bbb R};H)}=\sup\limits_{t\in{\Bbb R}}\int^{t+1}_{t}\|f(t)\|^{2}_{H}{\rm d}s.$记空间$L^{2}_{\rm loc}({\Bbb R};H)$中的平移紧函数构成的空间为$L^{2}_{c}({\Bbb R};H), $而$L^{2}_{c}({\Bbb R};H)=\{f\in L^{2}_{\rm loc}({\Bbb R};H):$任意的$[t_{1}, t_{2}]\subset{\Bbb R}, \{f(x, \tau+s):\tau\in{\Bbb R}\}\mid_{[t_{1}, t_{2}]}$在空间$ L^{2}(t_{1}, t_{2};H)$中是准紧的.令$L^{2}_{\rm loc}({\Bbb R};H)$的局部弱收敛空间为$L^{2}_{w, loc}({\Bbb R};H), $即空间$L^{2}_{w, loc}({\Bbb R};H)$中存在一个序列${f_{n}}\longrightarrow{f}(n\rightarrow\infty)$当且仅当

$\int^{t_{2}}_{t_{1}}\int_{\Omega}(f_{n}(x, s)-f(x, s))\varphi(x, s){\rm d}x{\rm d}s\rightarrow0, \;\;\; n\rightarrow\infty, $

对所有的 $[t_{1}, t_{2}]\subset{\Bbb R}$ 及$\varphi(x, s)\in L^{2}(t_{1}, t_{2};H)$成立.

我们引入空间$X$中一个集合$B_{1}$到另一个集合$B_{2}$的Hausdorff半距离,即

${\rm dist}_{X}(B_{1}, B_{2})=\sup\limits_{b_{1}\in{B_{1}}}\inf\limits_{b_{2}\in{B_{2}}}\|b_{1}-b_{2}\|. $

记$L^p(\Omega)$, $1<p<\infty$为Lebesgue空间, $H^{m}(\Omega)=H^{m, 2}(\Omega)$是Sobolev空间. $<\cdot, \cdot>$和$\|\cdot\|$分别表示$L^{2}(\Omega)$中的内积和范数.令$V_{0}=\{v\mid v\in (C_{0}^{\infty}(\Omega))^{3}\}, $定义$H$是$V_{0}$在$(C_{0}^{\infty}(\Omega))^{3}$中的闭包,记$V:={\cal H}_{0}=(H_{0}^{1}(\Omega))^{3}\bigcap H.$$P$为$(L^{2}(\Omega))^{3}$在$H$上的Helmholtz-Leray正交映射,则$A=-P\triangle$是$D(A)=(H^{2}(\Omega))^{3}\bigcap V$中的Stokes算子.考虑一族Hilbert空间$D(A^{\frac{s}{2}})$, $s\in{\Bbb R}, $其内积和范数分别如下表示

$\langle \cdot, \cdot\rangle_{D(A^{\frac{s}{2}})}=\langle A^{\frac{s}{2}}\cdot, A^{\frac{s}{2}}\cdot\rangle, \;\;\;\;\;\|\cdot\|_{D(A^{\frac{s}{2}})}=\|A^{\frac{s}{2}}\cdot\|.$

对任意的$s>r, $有$D(A^{\frac{s}{2}})\hookrightarrow{D(A^{\frac{r}{2}})}.$对$\forall u=(u_{1}, u_{2}, u_{3})$和$v=(v_{1}, v_{2}, v_{3}), $定义

$\langle u, v\rangle =\sum^3\limits_{j=1}\langle u_{j}, v_{j}\rangle_{L^{2}(\Omega)} , \;\;\;\|v\|^{2}=\sum^3\limits_{j=1}\|v_{j}\|_{L^{2}(\Omega)}^{2}, $

分别为空间$H$中的内积与范数.由Poincaré不等式得, $V$的等价范数为

$\|v\|^2_V=\|\nabla v\|^{2}=\sum^3\limits_{i, j=1}\|\partial_{i}v_{j}\|^2_{L^{2}(\Omega)}. $

定义$V_{1}:={\cal H}_{1}=({H}^{2}(\Omega))^{3}\cap{V}.$当$0\leq s\leq1$时,可定义如下的一族Hilbert空间

${\cal H}_{s}=D(A^{\frac{s}{2}}) ,\;\;\;\|\cdot\|_{{\cal H}_{s}}=\|\cdot\|_{s}.$

对非线性向量函数$\overrightarrow{F}=(F_{1}(s), F_{2}(s), F_{3}(s)), \forall s\in{\Bbb R}, $定义

(2.1) $G_{i}(s)=F'_{i}(s),\;\;\;{\cal F}_{i}(s)=\int^{s}_{0}F_{i}(r){\rm d}r, $

其中

(2.2) $\overrightarrow{G}(s)=(G_{1}(s), G_{2}(s), G_{3}(s)), \;\;\;\overrightarrow{{\cal F}}(s)=({\cal F}_{1}(s), {\cal F}_{2}(s), {\cal F}_{3}(s)).$

假设$F_{i}(i=1, 2, 3)$满足与文献[8]相同的条件,即$F_{i}$是光滑函数并且满足

(2.3) $F_{i}(0)=0, \ |F_{i}(s)|\leq c_{1}|s|+c_{2}|s|^{2},\;\;\;\forall s\in{\Bbb R},$

(2.4) $c_{1}(1+c_{0}|s|)\leq|G_{i}(s)|\leq c_{2}(1+c_{0}|s|), \ |{\cal F}_{i}(s)|\leq c_{1}|s|^{2}+c_{2}|s|^{3}, \;\;\;\forall s\in{\Bbb R},$

其中$c_{0}$, $c_{1}$和$c_{2}$是正常数.

另外,本文会经常用到以下Poincaré不等式

(2.5) $\lambda_{1}\|u\|^{2}\leq\|u\|^{2}_{V},\;\;\;\forall u\in V, $

这里$\lambda_{1}$是正常数.

定理2.1^[5]  设$({\cal M}, d)$是一个度量空间, $U(t, \tau)$是${\cal M}$中Lipschitz连续的解过程,满足

$d(U(t, \tau)m_{1}, U(t, \tau)m_{2})\leq C{\rm e}^{k(t-\tau)}d(m_{1}, m_{2}), $

其中常数$C$和$k$与$m_{1}$, $m_{2}$, $t$, $\tau$无关.进一步假设存在三个子集${\cal M}_{1}, {\cal M}_{2}, {\cal M}_{3}\subset{\cal M}$使得

${\rm dist}_{\cal M}(U(t, \tau){\cal M}_{1}, {\cal M}_{2})\leq L_{1}{\rm e}^{-v_{1}t}, \\{\rm dist}_{\cal M}(U(t, \tau){\cal M}_{2}, {\cal M}_{3})\leq L_{2}{\rm e}^{-v_{2}t}, $

这里$v_{1}, v_{2}>0$且$L_{1}, L_{2}>0.$则有

${\rm dist}_{\cal M}(U(t, \tau){\cal M}_{1}, {\cal M}_{3})\leq L{\rm e}^{-vt}, $

其中$v=\frac{v_{1}v_{2}}{k+v_{1}+v_{2}}, L=CL_{1}+L_{2}.$

本小节我们考虑自治的BBM方程,即下面的方程

(3.1) $\left\{\begin{array}{ll}u_{t}+Au_{t}+\nu Au+\nabla\cdot\overrightarrow{F}(u)=f(x), & (x, t)\in\Omega\times(0, \infty), \\u(x, t)|_{\partial\Omega}=0, \\u(x, 0)=u_{0}(x),&x\in\Omega, \end{array} \right.$

这里$f(x)=Pg(x)\in V', $其中$V'$是$V$的对偶空间.

定义3.1^[6] (加强的平坦性条件)  设$X$为一致凸的Banach空间,如果对任意的有界子集$B\subset X, $存在$X$的有限维子空间$X_{1}\subset X, $及$k, l>0$和$T>0, $使得

(1) $P_{m}(\bigcup\limits_{s\geq t}S(s)B)$有界;

(2) $\|(I-P_{m})(\bigcup\limits_{s\geq t}S(s)x)\|_{X}\leq k{\rm e}^{-lt}+\phi(m), \forall x\in B, t\geq T.$

其中, $P_{m}:X\longrightarrow X_{1}$为有界投影, dim$X_{1}=m, $$\phi(m)$为一实函数,满足$\lim\limits_{s\rightarrow\infty}\phi(s)=0.$则称此条件为加强的平坦性条件.

定理3.1^[8]  设$\{S(t)\}_{t\geq0}$为完备度量空间$X$中的半群, $B$为$\{S(t)\}_{t\geq0}$在空间$X$中的有界吸收集,则以下条件等价

(1) $\bigcup\limits_{s\geq t}S(s)B$的非紧性测度是指数衰退的,即存在$k, l>0$使得

(3.2) $\alpha(\bigcup\limits_{s\geq t}S(s)B)\leq k{\rm e}^{-lt}, $

其中$\alpha$表示Kuratowski非紧性测度.

(2)半群$\{S(t)\}_{t\geq0}$在$X$中拥有指数吸引子.

定理3.2^[6]  设$X$中的半群$\{S(t)\}_{t\geq0}$满足强拉平性,则$\bigcup\limits_{s\geq t}S(s)B$的非紧性测度是指数衰退的.

定理3.3^[6]  设$X$为一致凸的Banach空间, $\{S(t)\}_{t\geq0}$为$X$中的强连续或强弱连续半群,则$\{S(t)\}_{t\geq0}$在$X$中拥有指数吸引子,如果它满足

(1) $\{S(t)\}_{t\geq0}$在$X$中存在有界吸收集$B\subset X$;

(2) $\{S(t)\}_{t\geq0}$满足加强的平坦性条件.

定理3.4^[8]  设$f(x)\in V', $那么对于任意的$u_{0}\in V, $$T>0, $方程$(3.1)$存在唯一解$u$

(3.3) $u\in C([0, T];V)\cap L^{\infty}(0, \infty;V_{0}),\;\;\;u_{t}\in L^{2}(0, T;V).$

并且, $u$连续依赖于$V$中的初值.

定理3.5^[8]  存在正常数$\rho$和$T=T(B), $使得对任意的有界集$B\subset V$,有

(3.4) $\|S(t)u_{0}\|_{V}\leq \rho,\;\;\; t\geq T,\;\;\;\forall u_{0}\in B.$

下面证明半群$\{S(t)\}_{t\geq0}$在空间$V$中满足加强的平坦性条件.设$\lambda_{i}, i=1, 2, \cdot\cdot\cdot$为算子$A$在空间$V$中的特征值,满足$0<\lambda_{1}\leq\lambda_{2}\leq\cdots \leq\lambda_{j}\leq\cdots , \lambda_{j}\rightarrow\infty, j\rightarrow\infty, \omega_{i}$表示特征值$\lambda_{i}$对应的特征向量,它构成空间$V$中的正交基,并且满足

$A\omega_{i}=\lambda_{i}\omega_{i},\;\;\;\forall i\in N.$

设$V_{m}={\rm span}\{\omega_{1}, \omega_{2}, \cdots , \omega_{m}\}, P_{m}:V\longrightarrow V_{m}$为正交投影.对任意的$(u, u_{t})\in V, $记

$(u, u_{t})=(u_{1}, u_{1t})+(u_{2}, u_{2t}), $

其中$(u_{1}, u_{1t})=(P_{m}u, P_{m}u_{t}).$

定理3.6  设$\{S(t)\}_{t\geq0}$为方程$(3.1)$的解半群, $f(x)=Pg(x)\in H, $则半群$\{S(t)\}_{t\geq0}$在空间$V$中满足加强的平坦性条件,即对于任意有界集$B\subset V, $存在常数$k, l, T>0$和函数$q(m), $使得对任意的$Z_{0}\in B, $$t\geq T, $有

$\bigg\|(I-P_{m})\bigcup\limits_{s\geq t}S(s)Z_{0}\bigg\|^{2}\leq k{\rm e}^{-lt}+q(m), $

其中$\lim\limits_{m\rightarrow\infty}q(m)=0.$

证  用$u_{2}$作为试验函数与方程$(3.1)$在空间$H$中做内积,可得

(3.5) $\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|u_{2}\|^{2}+\|\nabla u_{2}\|^{2})+\nu\|\nabla u_{2}\|^{2}+\langle\nabla\overrightarrow{F}(u_{2}), u_{2}\rangle\leq\langle f(x), u_{2}\rangle , $

其中

(3.6) $\langle f(x), u_{2}\rangle=\frac{1}{2\nu}\|(I-P_{m})f(x)\|^{2}+\frac{\nu}{2}\|u_{2}\|^{2}.$

定义泛函

$L(t)=\frac{1}{2}(\|u_{2}\|^{2}+\|\nabla u_{2}\|^{2}), $

则结合$(3.3)$式, $(3.4)$式与$(3.6)$式可得

$\frac{\rm d}{{\rm d}t}L(t)+\nu L(t)\leq\|(I-P_{m})f(x)\|^{2},\;\;\; t\geq t_{0}, $

由Gronwall引理,可得

$L(t)\leq L(t_{0}){\rm e}^{-\nu(t-t_{0})}+\int_{t_{0}}^{t}{\rm e}^{-\nu(t-t_{0})}\|(I-P_{m})f(x)\|^{2}{\rm d}s,\;\;\; t\geq t_{0}. $

因此,方程$(3.1)$的解半群在空间$V$中满足加强的平坦性条件.

利用定理$3.4$即得下面的结论.

定理3.7  设$\{S(t)\}_{t\geq0}$为方程$(3.1)$的解半群, $f(x)=Pg(x)\in H, $则半群$\{S(t)\}_{t\geq0}$在空间$V$中拥有指数吸引子.

本小节我们考虑下面非自治的BBM方程

(4.1) $ \left\{\begin{array}{ll} u_{t}+Au_{t}+\nu Au+\nabla\cdot\overrightarrow{F}(u)=f(x, t), &(x, t)\in\Omega\times[\tau, \infty), \\ u(x, t)=0, &(x, t)\in\partial\Omega\times[\tau, \infty), \\ u(x, \tau)=u_{\tau}(x),&\tau\in{\Bbb R}.\end{array} \right.$

对外力项$f, $我们仅假设$f_{0}=f_{0}(x, t)\in L_{b}^{2}({\Bbb R};H), $并设$\Sigma_{0}=\{(x, t)\longrightarrow f_{0}(x, t+h):h\in{\Bbb R}\}.$在$L_{w, loc}^{2}({\Bbb R};H)$中,令$\Sigma={\cal H}(f_{0}), $它是$\Sigma_{0}$关于$L_{\rm loc}^{2}({\Bbb R};H)$的局部弱拓扑的闭包.

定理4.1^[8]  假设$f(x, t)\in L^{2}_{b}({\Bbb R};H), $那么对于任意的$\tau\in {\Bbb R}$和任意的初值$u_{\tau}\in V, $方程$(4.1)$存在唯一解.进一步对方程$(4.1)$的任意两个解$u_{1}(t)$和$u_{2}(t), $有

$\|u_{1}(t)-u_{2}(t)\|^{2}_{V}\leq Q(u_{\tau1}, u_{\tau2}, T)(\|u_{\tau1}-u_{\tau2}\|^{2}_{V}+\|f_{1}-f_{2}\|^{2}_{L^{2}_{b}({\Bbb R};H)}),\;\;\;\forall\tau\leq t\leq T, $

$\forall u_{\tau i}\in V$和$f_{i}\in L^{2}_{b}({\Bbb R};H), i=1, 2.$因此,过程${U_{f}(t, \tau)}:U_{f}(t, \tau)u_{\tau}=u(t), $和$U_{f}(t, \tau):V\longrightarrow V, $$t\geq\tau, \tau\in{\Bbb R}$是有意义的.这里$u(t)$是方程$(4.1)$的解.

定理4.2^[8]  设(2.3)-(2.4)式成立, $f\in L^{2}_{b}({\Bbb R}, H), $那么对任意的有界集$B\subset V, $存在一个依赖于$\|u_{\tau}\|_{V}$和$\|f\|_{L^{2}_{b}({\Bbb R}, H)}$的常数$M, $使得

(4.2) $\|U_{f}(t, \tau)u_{\tau}\|^{2}_{V}\leq M,\;\;\;\forall t\geq \tau,\;\;\;u_{\tau}\in B.$

为了得到解的正则性的估计,将$U_{f}(t, \tau){u_{\tau}}=u(t)$做如下分解

(4.3) $U_{f}(t, \tau){u_{\tau}}=D(t, \tau){u_{\tau}}+K_{f}(t, \tau){u_{\tau}}, $

其中$D(t, \tau)u_{\tau}=v(t), K_{f}(t, \tau)u_{\tau}=w(t)$分别是下面两个方程的解

(4.4) $ \left\{\begin{array}{ll} v_{t}+Av_{t}+\nu Av=0, &(x, t)\in\Omega\times[\tau, \infty), \\ v(x, t)=0, &(x, t)\in\partial\Omega\times[\tau, \infty), \\ v(x, \tau)=u_{\tau}(x),&x\in\Omega\end{array} \right.$

和

(4.5) $\left\{\begin{array}{ll} w_{t}+Aw_{t}+\nu Aw+\nabla\cdot\overrightarrow{F}(u)=f(t), &(x, t)\in\Omega\times[\tau, \infty), \\ w(x, t)=0, &(x, t)\in\partial\Omega\times[\tau, \infty), \\ w(x, \tau)=0, &x\in\Omega.\end{array} \right.$

定理4.3^[8]  对任意的$\tau\in{\Bbb R}$和初值$u_{\tau}\in {\cal H}_{0}, $方程$(4.4)$的解满足以下结果:存在$k>0$使得对任意的$t\geq\tau$有

$\|D(t, \tau)u_{\tau}\|_{{\cal H}_{0}}^{2}=\|v(t)\|_{V}^{2}\leq Q(\|u_{\tau}\|_{{\cal H}_{0}}){\rm e}^{-k(t-\tau)}, $

这里$Q(\cdot)$是非负的递增函数, $k$仅依赖于$\lambda_{1}.$

定理4.4^[8]  对任意的$T>0, f\in L_{b}^{2}({\Bbb R};H)$和$u_{\tau}\in V, $都存在一个与$T, \|f\|_{L_{b}^{2}({\Bbb R};H)}, \|u_{\tau}\|_{V}$有关的常数$M_{1}, $使得方程$(4.5)$的解满足

$\|K_{f}(T+\tau, \tau)u_{\tau}\|_{1+\sigma}^{2}=\|w(T+\tau)\|_{1+\sigma}^{2}\leq M_{1}, $

其中$0<\sigma<\frac{1}{2}.$

定理4.5^[8]  对$\forall\tau\in {\Bbb R}, $$u(t)$是方程$(4.1)$对应于初值$u_{\tau}\in{\cal H}_{0}$的解.则对$\forall\epsilon>0, $分解$u(t)=v_{1}(t)+w_{1}(t), \forall t\geq\tau, $其中$v_{1}(t)$和$w_{1}(t)$分别满足

$\|A^{\frac{1+\sigma}{2}}w_{1}(t)\|^{2}\leq K_{\epsilon},\;\;\;\forall t\geq\tau$

和

$\int_{s}^{t}\|\nabla v_{1}(r)\|^{2}{\rm d}r\leq\epsilon(t-s)+C_{\epsilon},\;\;\;\forall t\geq s\geq\tau, $

其中$C_{\varepsilon}$和$K_{\epsilon}$是依赖于$\epsilon, \|u_{\tau}\|_{{\cal H}_{0}}$和$\|f\|_{L_{b}^{2}}^{2}({\Bbb R};H)$的常数.

定理4.6^[8]  设$B_{\theta}\subset{\cal H}_{\theta}$是有界的,那么对任意的$\theta\in[\sigma, 1-\sigma]$和$\tau\in{\Bbb R}$有

$\|K_{f}(t, \tau)u_{\tau}\|_{{\cal H}_{\theta+\sigma}}\leq K_{\theta}, t\geq\tau,\;\;\;\forall u_{\tau}\in B_{\theta}, $

其中常数$K_{\theta}$仅依赖于有界集$B_{\theta}$的${\cal H}_{\theta}$范数.

定理4.7^[8] (一致吸引子)  方程$(4.1)$在${\cal H}_{0}$中有一致吸引子${\cal A}, $并且${\cal A}$在${\cal H}_{1}$中有界,满足

${\cal A}=\bigcup\limits_{f\in\sum}{\cal K}_{f}(0), $

其中${\cal K}_{f}$是过程$U_{f}$的核, ${\cal K}_{f}(0)$是$0$时刻的核截片.

下面,我们研究问题(4.1)的非自治的指数吸引子.首先需要下面的一些抽象结果.

定义4.1^[5] (指数吸引子)  过程族$U(t, \tau), t\geq\tau, t\geq\tau$的紧的正不变集,即$(U(t, \tau){\cal M}(\tau)\subset{\cal M}(t))$的一个映射$t\longrightarrow{\cal M}(t)$,在Banach空间$E$中称为指数吸引子,如果以下条件成立

(1)所有集合${\cal M}$的分形维数是适定的并且一致有界

${\rm dim}_{F}({\cal M}(t), E)\leq C < \infty,\;\;\;t\in{\Bbb R};$

(2)存在一个正常数$\alpha$和一个单调函数$Q, $使得对每一个$t\in{\Bbb R}, s\geq0$和每一个在$E$中的有界集$B, $都有

${\rm dist}_{E}(U(t+s, t)B, {\cal M}(t+s))\leq Q(\|B\|_{E}){\rm e}^{-\alpha s}.$

定义4.2^{[5, 15]}  设$E$和$E_{1}$是两个Banach空间, $E_{1}\hookrightarrow E$, $B$是$E_{1}$的一个有界吸收集.给定三个常数$\varepsilon\in[0, 1)$和$\delta, K>0, $${\Bbb S}_{\delta, \varepsilon, K}(B)$为从$E$到其自身的一簇非线性算子.我们说$S\in{\Bbb S}_{\delta, \varepsilon, K}(B), $如果它满足

(1)映射$S$将$B$中的$\delta$-邻域${\cal O}_{\delta}(B)$映射到$B$中

$S:{\cal O}_{\delta}(B)\longrightarrow B;$

(2) $S$有以下的分解形式

$S=S_{0}+S_{1}, S_{0}:{\cal O}_{\delta}\longrightarrow E, S_{1}:{\cal O}_{\delta}\longrightarrow E_{1}, $

这里$S_{0}$和$S_{1}$分别满足以下条件

$\|S_{0}(z_{1})-S_{0}(z_{2})\|_{E}\leq\varepsilon\|z_{1}-z_{2}\|_{E},\;\;\;\forall z_{1}, z_{2}\in{\cal O}_{\delta}(B)$

和

(4.6) $\|S_{1}(z_{1})-S_{1}(z_{2})\|_{E_{1}}\leq K\|z_{1}-z_{2}\|_{E},\;\;\;\forall z_{1}, z_{2}\in{\cal O}_{\delta}(B).$

定理4.8^{[5, 15]}  设$E$和$E_{1}$是两个Banach空间, $E_{1}$紧嵌入到$ E, $$B$是$E_{1}$中的一个有界子集并且$U(t, \tau), t\geq\tau, \tau\in{\Bbb R}$是$E$上的过程.进一步假设过程$U(t, \tau)$满足以下条件

(a)存在正常数$\varepsilon<1, \delta, K, $使得:存在一个$T>0$,使得$U(T+\tau, \tau)\in{\Bbb S}_{\delta, \varepsilon, K}(B)$对任意的$\tau\in{\Bbb R}$都成立;

(b) $U(t, \tau)$依下面的意义下是Hölder连续的：存在正常数$k_{1}$和$C_{T, B}, $使得对任意的$z_{i}\in{\cal O}_{\delta} (B), i=1, 2, s\in[0, T], $$\tau\in{\Bbb R}, $有

(4.7) $\|U(\tau+s+t, \tau)z_{1}-U(\tau+t, \tau)z_{1}\|_{E}\leq C_{T, B}|s|^{1/2},\;\;\;\forall t \geq 0$

和

(4.8) $\|U(\tau+s+t, \tau+s)z_{2}-U(\tau+t, \tau)z_{2}\|_{E}\leq C_{T, B}{\rm e}^{ct}|s|^{k_{1}}, \;\;\;\forall t\geq T.$

那么,存在一个指数吸引子$\tau\rightarrow\varepsilon_{U}(\tau)\subset{\cal O}_{\delta}(B), \tau\in{\Bbb R}, $满足以下性质

(1)对每一个$\tau\in{\Bbb R}, $$\varepsilon_{U}(\tau)\subset B$并且它的分形维数是有限的

(4.9) ${\rm dim}_{F}(\varepsilon_{U}(\tau), E)\leq C_{1}, $

这里常数$C_{1}$与$\tau$无关;

(2) $\varepsilon_{U}(\tau)$关于$U$是正不变的

(4.10) $U(t, \tau)\varepsilon_{U}(\tau)\subset\varepsilon_{U}(t),\;\;\;t, \tau\in{\Bbb R}, t\geq\tau;$

(3)这一簇集合满足下列形式的一致指数吸引性

(4.11) ${\rm dist}_{E}(U(t, \tau)B, \varepsilon_{U}(t))\leq C_{2}{\rm e}^{-\alpha(t-\tau)}, \;\;\; \forall t\geq\tau, $

这里$C_{2}$和$\alpha$是与$\tau, t$无关的正常数;

(4)函数$t\longrightarrow\varepsilon_{U}(t)$是一致Hölder连续的

(4.12) ${\rm dist}^{symm}_{E}(\varepsilon_{U}(t+s), \varepsilon_{U}(t))\leq C_{3}|s|^{k},\;\;\;\forall t\in{\Bbb R}, $

这里$C_{3}$和$k$是与$s, t$无关的正常数.

进一步我们假设$\overrightarrow{F}(u)$满足(2.3)-(2.4)式, $f\in L^{2}_{b}({\Bbb R};H)$并且有

(4.13) $\|f\|^{2}_{L^{2}_{b}({\Bbb R};H)}\leq M'<\infty.$

本文的主要结果是

定理4.9  设$\Omega$是${\Bbb R}^{3}$中的光滑有界域,且(2.3)-(2.4)式成立.那么当外力项$f$满足$(4.13)$式时,过程$U_{f}(t, \tau)$在${\cal H}_{0}$中存在非自治的指数吸引子$\{\varepsilon_{f}(t)\}_{t\in{\Bbb R}}, $并满足以下性质

(1)对每个$t\in{\Bbb R}, $$\varepsilon_{f}(t)$在${\cal H}_{0}$中是紧的,其分形维数是有限的,即

(4.14) ${\rm dim}_{F}(\varepsilon_{f}(t), {\cal H}_{0})\leq C_{1},\;\;\;t\in{\Bbb R}, $

(2) $\varepsilon_{f}(\tau)$关于$U_{f}$是正不变的,即

(4.15) $U_{f}(t, \tau)\varepsilon_{f}(\tau)\subset\varepsilon_{f}(t) \;\;\; t, \tau\in{\Bbb R}, t\geq\tau;$

(3) $\varepsilon_{f}(t)$满足一致指数吸引性:存在一个正常数$\alpha$和单调函数$Q(\cdot)$ (两者仅依赖于$\|f\|_{L^{2}_{b}({\Bbb R}, L^{2}(\Omega))}), $使得对${\cal H}_{0}$中的任意有界集$B, $当$t\geq\tau$时,有

(4.16) ${\rm dist}_{{\cal H}_{0}}(U_{f}(t, \tau)B, \varepsilon_{f}(t))\leq Q(\|B\|_{{\cal H}_{0}}){\rm e}^{-\alpha(t-\tau)}, $

这里常数$\alpha$和$Q(\cdot)$与$t, \tau$无关;

(4)函数$t\longrightarrow\varepsilon_{f}(t)$是Hölder连续的:对任意的$t\in{\Bbb R}, $有

(4.17) ${\rm dist}^{symm}_{{\cal H}_{0}}(\varepsilon_{f}(t+s), \varepsilon_{f}(t))\leq C_{2}|s|^{k}, $

这里常数$C_{2}$和$k$与$t, s$无关.

定理4.10$({\cal H}_{1}$中的一致耗散性)  设$\frac{1}{4}<\sigma<\frac{1}{2}, $存在一个仅与$\|f\|_{L^{2}_{b}({\Bbb R}, H)}$相关的正数${\Bbb M}, $使得对任意的有界(在${\cal H}_{1})$集$B\subset{\cal H}_{1}, $存在$T=T(\|B\|_{{\cal H}_{1}})>0$使得

(4.18) $\|U_{f}(t, \tau)u_{\tau}\|^{2}_{{\cal H}_{1}}\leq {\Bbb M}, \;\;\; t-\tau\geq T,\;\;\;\forall u_{\tau}\in B.$

证  由定理$4.2$可得存在一个依赖于有界集$B$的界的$T_{B}$使得

(4.19) $\|U_{f}(t, \tau)u_{\tau}\|_{{\cal H}_{0}}^{2}\leq M_{0},\;\;\; t-\tau\geq T_{B},\;\;\;\forall u_{\tau}\in B.$

由文献[8,定理4.6]可得,对任意的有界集$B\subset{\cal H}_{0}$和$\frac{1}4{}<\sigma<\frac{1}{2}, $存在常数$k_{\sigma}=k_{\sigma}$$(\|B\|_{{\cal H}_{0}}), $使得对任意$\tau\in{\Bbb R}, $有

(4.20) $\|K_{f}(t, \tau)u_{\tau}\|_{1+\sigma}^{2}\leq k_{\sigma},\;\;\;t\geq\tau,\;\;\;\forall u_{\tau}\in \bigcup\limits_{s\in{\Bbb R}}U_{f}(s+T_{B}, s)B.$

用$-\bigtriangleup u(t)$与方程$(4.1)$做内积得

(4.21) $\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\nabla u\|^{2}+\|\triangle u\|^{2})+\nu\|\triangle u\|^{2}-\int_{\Omega}\nabla\cdot\overrightarrow{F}(u)\triangle u{\rm d}x=-\int_{\Omega}f(x, t)\triangle u{\rm d}x.$

由Young不等式可得

(4.22) $\bigg|\int_{\Omega}f(x, t)\triangle u{\rm d}x\bigg|\leq \frac{2}{\nu}\|f(x, t)\|^{2}+\frac{\nu}{8}\|\triangle u(t)\|^{2}.$

结合$(2.3)$式, $(2.4)$式和$(4.19)$式可知

$\begin{eqnarray*}\int_{\Omega}\triangle u\nabla\cdot\overrightarrow{F}(u){\rm d}x&=&-\int_{\Omega}\nabla u{\rm d}(\overrightarrow{F}'(u)\nabla u)\\&=&\int_{\Omega}\nabla\cdot\overrightarrow{F}(u){\rm d}\nabla u\\&=&\nabla\cdot\overrightarrow{F}(u)\nabla u\mid_{\partial\Omega}-\int_{\Omega}\nabla u{\rm d}(\overrightarrow{F}'(u)\nabla u)\\&=&-\int_{\Omega}\bigtriangledown u\overrightarrow{F}^{''}(u)(\nabla u)^{2}{\rm d}x-\int_{\Omega}\nabla u\overrightarrow{F}'(u)(\triangle u){\rm d}x\\&\leq& C\int_{\Omega}|\nabla u|^{3}{\rm d}x+c_{2}\int_{\Omega}\nabla u(1+c_{0}|u|)\triangle u{\rm d}x\\&\leq& M_{1}+c_{2}\int_{\Omega}|\nabla u||\triangle u|{\rm d}x+c_{2}c_{0}\int_{\Omega}|\nabla u||u||\triangle u|{\rm d}x, \end{eqnarray*}$

由$D(A^{\frac{1+\sigma}{2}})\hookrightarrow L^{\frac{6}{1-2\sigma}}(\Omega), $$ D(A^{\frac{1-2\sigma}{4}})\hookrightarrow L^{\frac{6}{2+2\sigma}}(\Omega) $和$ D(A^{\frac{3-2\sigma}{4}})\hookrightarrow D(A^{\frac{1-2\sigma}{4}})$可得

$\begin{eqnarray*}\int_{\Omega}|\nabla u||u||\triangle u|{\rm d}x&\leq&\int_{\Omega}|v_{1}||\nabla u||\triangle u|{\rm d}x+\int_{\Omega}|w_{1}||\nabla u||\triangle u|{\rm d}x \\&\leq&\lambda_{1}\|\nabla v_{1}\|^{2}\|\triangle u\|^{2}\\&&+\bigg(\int_{\Omega}|w_{1}|^{\frac{6}{1-2\sigma}}\bigg)^{\frac{1-2\sigma}{6}}\bigg(\int_{\Omega}|\nabla u|^{\frac{6}{2+2\sigma}}\bigg)^{\frac{2+2\sigma}{6}}\bigg(\int_{\Omega}|\triangle u|^{2}\bigg)^{\frac{1}{2}}\\&\leq&\lambda_{1}\|\nabla v_{1}\|^{2}\|\triangle u\|^{2}+\|w_{1}\|_{L^{\frac{6}{1-2\sigma}}}\|\nabla u\|_{L^{\frac{6}{2+2\sigma}}}\|\triangle u\|\\&\leq&\lambda_{1}\|\nabla v_{1}\|^{2}\|\triangle u\|^{2}+\|A^{\frac{1+\sigma}{2}}w_{1}\|\|A^{\frac{1+\sigma}{2}}u\|\|\triangle u\|\\&\leq&\lambda_{1}\|\nabla v_{1}\|^{2}\|\triangle u\|^{2}+k_{\epsilon\sigma}\|\triangle u\|\\&\leq&\lambda_{1}\|\nabla v_{1}\|^{2}\|\triangle u\|^{2}+\frac{\nu}{8}\|\triangle u\|^{2}+\frac{2k_{\varepsilon\sigma}}{\nu}, \end{eqnarray*}$

$\int_{\Omega}|\nabla u||\triangle u|{\rm d}x\leq\bigg (\int_{\Omega}|\nabla u|^{2}\bigg)^{\frac{1}{2}}\bigg(\int_{\Omega}|\triangle u|^{2}\bigg)^{\frac{1}{2}}\leq\frac{\nu}{8}\|\triangle u\|^{2}+\frac{2c_{2}}{\nu}, $

故可得

(4.23) $\int_{\Omega}\nabla\overrightarrow{F}\triangle u{\rm d}x\leq\lambda_{1}\|\nabla v_{1}\|^{2}\|\triangle u\|^{2}+\frac{\nu}{4}\|\triangle u\|^{2}+\frac{k_{\varepsilon\sigma}}{2}+\frac{2c_{2}}{\nu}+M_{1} .$

将$(4.22)$和$(4.23)$式带入$(4.21)$式,当$t-\tau\geq T_{B}$并且$u_{\tau}\in B$时有

(4.24) $\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\nabla u\|^{2}+\nu\|\triangle u\|^{2})+(\frac{3}{8}\nu-\lambda_{1}\|\nabla v_{1}\|^{2})\|\triangle u(t)\|^{2}\leq \frac{2}{\nu}\|f(x, t)\|^{2}+\frac{k_{\varepsilon\sigma}}{2}+\frac{2c_{2}}{\nu}+M_{1} .$

利用Gronwall不等式,结合定理$4.5$,即可证明$(4.18)$式成立.

为了得到以下的引理,记

(4.25) ${\Bbb B}=\{u\in{\cal H}_{1}:\|\triangle u\|^{2}\leq{\Bbb M}\}$

和

(4.26) ${\cal O}_{1}({\Bbb B})=\{u\in{\cal H}_{1}:\|\triangle u\|^{2}\leq{\Bbb M}+1\}.$

定理4.11$({\cal H}_{0}$中的Lipschitz连续性)  在定理$4.1$的假设下,对任意固定的时间$T>0, $过程$U_{f}(t, \tau)$在以下意义下是Hölder连续的:存在正常数$C_{T, {\Bbb B}}$和$c'$使得对任意的$z_{i}\in{\cal O}_{1}({\Bbb B}), i=1, 2, s\in[0, T], $有

(4.27) $\|U_{f}(\tau+s+t, \tau)z_{1}-U_{f}(\tau+t, \tau)_{z_{1}}\|_{{\cal H}_{0}}\leq C_{T, {\Bbb B}}|s|^{1/2},\;\;\; \forall t \geq 0$

和

(4.28) $\|U_{f}(\tau+s+t, \tau+s)z_{2}-U_{f}(\tau+t, \tau)_{z_{2}}\|_{{\cal H}_{0}}\leq C_{T, {\Bbb B}}{\rm e}^{c{'}t}|s|^{1/2},\;\;\; \forall t\geq T.$

证  用$u_{t}$与方程$(4.1)$做内积可得

(4.29) $\|u_{t}\|^{2}+\|\nabla u_{t}\|^{2}+\frac{\nu}{2}\frac{\rm d}{{\rm d}t}\|\nabla u\|^{2}+(\nabla\cdot\overrightarrow{F}(u), u_{t})=(f(x, t), u_{t}).$

$\begin{eqnarray*}\bigg|\int_{\Omega}(\nabla\cdot\overrightarrow{F}(u))u_{t}{\rm d}x\bigg|&=&\bigg|\int_{\Omega}\overrightarrow{F}\cdot\nabla u_{t}{\rm d}x\bigg|\\&\leq& c_{1}\int_{\Omega}|u||\nabla u_{t}|{\rm d}x+c_{2}\int_{\Omega}|u|^{2}|\nabla u_{t}|{\rm d}x\\&\leq&\frac{1}{2}\|\nabla u_{t}\|^{2}+c_{1}^{2}\|u\|+cc_{2}^{2}\|u\|_{{\cal H}_{0}}^{4}.\end{eqnarray*}$

$(f(x, t), u_{t})\leq\frac{1}{2}\|f\|_{L_{b}^{2}({\Bbb R};H)}^{2}+\frac{1}{2}\|u_{t}\|^{2}, $

因为$H_{0}^{1}(\Omega)\hookrightarrow L^{p}(\Omega) (p<6), $故可得

$\|u_{t}\|^{2}+\|\nabla u_{t}\|^{2}+\nu\frac{\rm d}{{\rm d}t}\|\nabla u\|^{2}\leq\frac{1}{2}\|f\|_{L_{b}^{2}({\Bbb R};H)}^{2}+{\lambda_{1}}\|u\|_{{\cal H}_{0}}^{2}+2cc_{2}^{2}\|u\|_{{\cal H}_{0}}^{4}.$

由$u(t)=U_{g}(t, \tau)u_{\tau} (t\geq\tau)$对任意的$u_{\tau}\in{\cal O}_{1}({\Bbb B})$将上式从[t, t+s]上积分可得

(4.30) $\int_{t}^{t+s}\|\nabla u_{t}(y)\|^{2}\leq|s|\|f\|_{L_{b}^{2}({\Bbb R};H)}^{2}+C_{M}.$

从而,利用与文献[5,定理$4.6$]同样的计算可得$(4.27)$式和$(4.28)$式成立.

定理4.12  在定理$4.1$的假设下,存在一个${\Bbb T}={\Bbb T}(B), $使得过程$U_{f}(t, \tau), t\geq\tau, $满足以下性质

(1)对任意的$\tau\in{\Bbb R}$和$t\geq0, $有

(4.31) $U_{f}(t+{\Bbb T}+\tau, \tau){\cal O}_{1}({\Bbb B})\subset{\Bbb B};$

(2) $\forall z_{1}, z_{2}\in{\cal O}_{1}({\Bbb B}), $对任意的$\tau\in{\Bbb R}, U_{f}({\Bbb T}+\tau, \tau)$做如下分解

$U_{f}({\Bbb T}+\tau, \tau)=S^{\tau}_{0}+S^{\tau}_{1}, \;\;\;S^{\tau}_{0}:{\cal O}_{1}({\Bbb B})\longrightarrow{\cal H}_{0}, \;\;\;S^{\tau}_{1}:{\cal O}_{1}({\Bbb B})\longrightarrow{\cal H}_{1}, $

这里$S^{\tau}_{0}$和$S^{\tau}_{1}$满足以下条件

(4.32) $\|S^{\tau}_{0}(z_{1})-S^{\tau}_{0}(z_{2})\|_{{\cal H}_{0}}\leq\frac{1}{4}\|z_{1}-z_{2}\|_{{\cal H}_{0}}, \;\;\;\forall z_{1}, z_{2}\in{\cal O}_{1}({\Bbb B})$

和

(4.33) $\|S^{\tau}_{1}(z_{1})-S^{\tau}_{1}(z_{2})\|_{{\cal H}_{1}}\leq C_{{\Bbb B}, {\Bbb T}}\|z_{1}-z_{2}\|_{{\cal H}_{0}}, \;\;\;\forall z_{1}, z_{2}\in{\cal O}_{1}({\Bbb B}).$

证  由定理$4.6$和$4.9, $对每个初值$u_{\tau}\in{\cal O}_{1}({\Bbb B}), $我们做如下分解:定义$S(t, \tau)u_{\tau}$为齐次方程$(4.1)$的解,并且$\widetilde{S}_{f}(t, \tau)u_{\tau}=U_{f}(t, \tau)u_{\tau}-S(t, \tau)u_{\tau}.$那么,对任意的两个初值$u^{i}_{\tau}\in{\cal O}_{1}({\Bbb B})$与其对应的两个解$u^{i}(t)$$=U_{f}(t, \tau)u^{i}_{\tau}, i=1, 2, $集合$u_{\tau}=u^{1}_{\tau}-u^{2}_{\tau}, $将$u^{1}(t)-u^{2}(t)$写成如下形式的和

$U_{f}(t, \tau)u^{1}_{\tau}-U_{f}(t, \tau)u^{2}_{\tau}=S_{f}(t, \tau)u^{1}_{\tau}-S_{f}(t, \tau)u^{2}_{\tau}+\widetilde{S}_{f}(t, \tau)u^{1}_{\tau}-\widetilde{S}_{f}(t, \tau)u^{2}_{\tau}, $

这里$\widetilde{v}(t)=S(t, \tau)u^{1}_{\tau}-S(t, \tau)u^{2}_{\tau}$是下列线性方程的解

(4.34) $\left\{\begin{array}{ll}\widetilde{v}(t)-\triangle\widetilde{v}(t)-\nu\triangle\widetilde{v}=0, \\\widetilde{v}(\tau, x)=u^{1}_{\tau}-u^{2}_{\tau}.\end{array} \right.$

而$\widetilde{w}(t)=\widetilde{S}_{f}(t, \tau)u^{1}_{\tau}-\widetilde{S}_{f}(t, \tau)u^{2}_{\tau}$是下列方程的解

(4.35) $\left\{\begin{array}{ll}\widetilde{w}(t)-\triangle\widetilde{w}(t)-\nu\triangle\widetilde{w}=\nabla\overrightarrow{F}(u^{1}(t))-\nabla\overrightarrow{F}(u^{2}(t)), \\\widetilde{w}(x)=0.\end{array} \right.$

由$(4.34)$式可得

$\|S(t, \tau)u^{1}_{\tau}-S(t, \tau)u^{2}_{\tau}\|_{{\cal H}_{0}}=\|\widetilde{v}_{t}\|_{{\cal H}_{0}}\leq C_{1}\|u^{1}_{\tau}-u^{2}_{\tau}\|_{{\cal H}_{0}}{\rm e}^{-k_{1}(t-\tau)}, $

这里常数$C_{1}$和$k_{1}$仅仅与第一特征值$\lambda_{1}$有关.因此,当$t-\tau$足够大时定义$T_{1}$有

(4.36) $\|S(t+T_{1}+\tau, \tau)u^{1}_{\tau}-S(t+T_{1}+\tau, \tau)u^{2}_{\tau}\|_{{\cal H}_{0}}\leq\frac{1}{4}\|u^{1}_{\tau}-u^{2}_{\tau}\|_{{\cal H}_{0}}, t\geq0.$

对于$\widetilde{w}(t), $用$-\triangle\widetilde{w}(t)$与$(4.35)$式做内积可得

(4.37) $\begin{array}[b]{ll}&\frac{1}{2}\frac{\rm d}{{\rm d}t}(\|\nabla\widetilde{w}(t)\|^{2}+\|\triangle\widetilde{w}(t)\|^{2})+\nu\|\triangle\widetilde{w}(t)\|^{2}\\\leq&\frac{1}{2\nu}\int_{\Omega}|\nabla\overrightarrow{F}(u^{1}(t))-\nabla\overrightarrow{F}(u^{2}(t))|^{2}{\rm d}x+\frac{\nu}{2}\|\triangle\widetilde{w}(t)\|^{2}{\rm d}x.\end{array}$

由Lagrange中值定理与Hölder不等式可得

$\begin{eqnarray*}&&\int_{\Omega}|\nabla\overrightarrow{F}(u^{1}(t))-\nabla\overrightarrow{F}(u^{2}(t))|^{2}{\rm d}x\\&=&\int_{\Omega}|\overrightarrow{F}'(u^{1}(t))\nabla u^{1}(t)-\overrightarrow{F}'(u^{2}(t))\nabla u^{2}(t)|^{2}{\rm d}x\\&=&\int_{\Omega}|\overrightarrow{F}'(u^{1}(t))(\nabla u^{1}(t)-\nabla u^{2}(t))+(\overrightarrow{F}'(u^{1}(t))-\overrightarrow{F}'(u^{2}(t)))\nabla u^{2}(t)|^{2}{\rm d}x\\&=&\int_{\Omega}|\overrightarrow{F}'(u^{1}(t))(\nabla u^{1}(t)-\nabla u^{2}(t))+\overrightarrow{F}^{''}(\xi)(u^{1}(t)-u^{2}(t))\nabla u^{2}(t)|^{2}{\rm d}x\\&\leq&\int_{\Omega}|\overrightarrow{F}'(u^{1}(t))(\nabla u^{1}(t)-\nabla u^{2}(t))|^{2}{\rm d}x+|\overrightarrow{F}^{''}(\xi)(u^{1}(t)-u^{2}(t))\nabla u^{2}(t)|^{2}{\rm d}x, \end{eqnarray*}$

其中$\xi=u^{1}(t)+\theta(u^{2}(t)-u^{1}(t)), \theta\in[0, 1].$

由$(2.4)$式与Hölder不等式可知

$\begin{eqnarray*}&&\int_{\Omega}|\nabla\overrightarrow{F}(u^{1}(t))-\nabla\overrightarrow{F}(u^{2}(t))|^{2}{\rm d}x\\&=&\int_{\Omega}|\overrightarrow{F}'(u^{1}(t))\nabla u^{1}(t)-\overrightarrow{F}'(u^{2}(t))\nabla u^{2}(t)|^{2}{\rm d}x\\&=&\int_{\Omega}|\overrightarrow{F}'(u^{1}(t))(\nabla u^{1}(t)-\nabla u^{2}(t))+(\overrightarrow{F}'(u^{1}(t))-\overrightarrow{F}'(u^{2}(t)))\nabla u^{2}(t)|^{2}{\rm d}x\\&=&\int_{\Omega}|\overrightarrow{F}'(u^{1}(t))(\nabla u^{1}(t)-\nabla u^{2}(t))+\overrightarrow{F}^{''}(\xi)(u^{1}(t)-u^{2}(t))\nabla u^{2}(t)|^{2}{\rm d}x\\&\leq&\int_{\Omega}|\overrightarrow{F}'(u^{1}(t))(\nabla u^{1}(t)-\nabla u^{2}(t))|^{2}{\rm d}x+|\overrightarrow{F}^{''}(\xi)(u^{1}(t)-u^{2}(t))\nabla u^{2}(t)|^{2}{\rm d}x, \end{eqnarray*}$

$\begin{eqnarray*}\int_{\Omega}|\overrightarrow{F}^{''}(\xi)|^{2}|u^{1}(t)-u^{2}(t)|^{2}|\nabla u(t)|^{4}{\rm d}x&\leq &C\int_{\Omega}|u^{1}(t)-u^{2}(t)|^{2}{\rm d}x\\&\leq &C\|u^{1}(t)-u^{2}(t)\|^{2}.\end{eqnarray*}$

故可得

$\int_{\Omega}|\nabla\overrightarrow{F}(u^{1}(t))-\nabla\overrightarrow{F}(u^{2}(t))|^{2}{\rm d}x\leq +C[ \|\nabla u^{1}(t)-\nabla u^{2}(t)\|^{2}+\|u^{1}(t)-u^{2}(t)\|^{2} ].$

从而,由$(4.37)$式可得

$\frac{\rm d}{{\rm d}t}(\|\nabla\widetilde{w}(t)\|^{2}+\|\triangle\widetilde{x}(t)\|^{2})\leq\frac{c}{2\nu}[ \|\nabla u^{1}(t)-\nabla u^{2}(t)\|^{2}+\|u^{1}(t)-u^{2}(t)\|^{2} ].$

因此,由定理$4.1$与定理$4.4$可得:对任意的$T>0$有

(4.38) $\|\nabla\widetilde{w}(\tau+T)\|^{2}+\|\triangle\widetilde{w}(\tau+T)\|^{2}\leq C_{M_{{\Bbb B}, T}}[ \|\nabla u^{1}(t)-\nabla u^{2}(t)\|^{2}+\|u^{1}(t)-u^{2}(t)\|^{2} ], $

其中$C_{M_{{\Bbb B}, T}}$是与$\tau$无关的常数.

故有

$\|\triangle\widetilde{w}(t)\|^{2}\leq C\|\nabla u^{1}(t)-\nabla u^{2}(t)\|^{2}.$

由定理$4.9$对于有界集${\cal O}_{1}({\Bbb B}), $存在一个仅依赖于${\cal H}_{1}$中的有界集${\Bbb B}$对任意的$\tau\in{\Bbb R}, $有

(4.39) $U_{f}(t, \tau){\cal O}_{1}({\Bbb B})\subset{\Bbb B}, t-\tau\geq T_{2}.$

取${\Bbb T}=\max\{T_{1}, T_{2}\}$和$S^{\tau}_{0}=S(\tau+{\Bbb T}, \tau), S^{\tau}_{1}=\widetilde{S}_{f}(\tau+{\Bbb T}, \tau).$那么从$(4.36)$式, $(4.38)$式与$(4.39)$式可知${\Bbb B}, S^{\tau}_{0}$和$S^{\tau}_{1}$满足条件(4.31)-(4.33)式.

定理4.9的证明  由定理$4.10$和$4.11, $再通过定理$4.8$可得对于$(4.25)$式中定义的集合${\Bbb B}$中外力项$f$满足条件$(4.13)$式时,集合${\Bbb B}$存在一个满足(4.14)-(4.17)式的非自治的拉回指数吸引子$\{\varepsilon_{f}(t)\}_{t\in{\Bbb R}}.$