本文讨论下面非自治格点长波-短波共振方程组的初值问题

\begin{equation} {\rm i}\dot{u}_{m}-(Au)_{m}-u_{m}v_{m}+{\rm i}\alpha u_{m}=f_{m}(t),\ m\in {\bf {\Bbb Z}},t >\tau,\label{1.1} \end{equation}

(1.1)

\begin{equation} \dot{v}_{m}+\beta v_{m}+\gamma(B(|u|^{2}))_{m} =g_{m}(t),\ m\in {\bf {\Bbb Z}},t >\tau,\label{1.2} \end{equation}

(1.2)

\begin{equation} u_{m}(\tau)=u_{m,\tau},\quad v_{m}(\tau)=v_{m,\tau}, \ m\in {\bf {\Bbb Z}},\tau\in{\bf {\Bbb R}},\label{1.3} \end{equation}

(1.3)

其中 $u_{m}(t)\in {\mathbb C}$(复数集),$v_{m}(t)\in {\mathbb R}$(实数集), ${\Bbb Z}$为整数集, i为虚数单位,$\alpha,\beta, \gamma$为正的常数, $A$ 和 $B $ 均为线性算子,分别定义为

\begin{equation} (Au)_{m} =2u_{m}-u_{m+1}-u_{m-1},\forall u=(u_{m})_{m\in {\bf {\Bbb Z}}},\label{1.4} \end{equation}

(1.4)

\begin{equation} (Bu)_{m}=u_{m+1}-u_{m} ,\forall u=(u_{m})_{m\in {\bf {\Bbb Z}}}.\label{1.5} \end{equation}

(1.5)

格点系统是某些变量离散化的时空系统,包括耦合的常微分方程组、耦合映射 格点和细胞自动机^{[12, 13]}. 在某些情况下,格点系统表现为偏微分方程的空间变量离散化近似. 格点系统在许多领域有广泛的应用,涉及电子工程^[11]、图象处理与模式识别^{[14, 15, 16]}、激光理论^[20]、材料科学^[22]、化学反应理论^{[19, 32]}、生物学^[31]等.

目前已有很多文献研究了格点系统. 例如,文献 [9, 26, 27, 28, 29] 研究了 随机格点系统,文献 [1, 2, 25, 40, 43, 44, 45, 46, 47, 49, 50]研究了格点系统指数吸引子,拉回指数吸引子 和一致指数吸引子,文献 [8, 17, 23, 24]研究了格点方程组的行波解问题. 同时,数学物理学家 们把格点动力系统的理论应用到许多数学物理模型中. 例如,文献 [7] 研究了格点 反应扩散方程组,文献 [30] 研究了三分子可逆 Gray-Scott 方程组,文献 [33]研究了离散的非线性 Schrödinger 方程组,文献 [36, 42] 研究了离散的耦合非线性 Schrödinger型方程组,文献 [51]研究了格点KGS方程组,文献 [52]研究了格点长波-短波共振方程组.

方程组 (1.1)-(1.2) 可以看作是下面非自治 长波-短波共振方程组在 ${\Bbb R}$ 上的一个离散近似

\begin{equation} {\rm i}u_{t}+u_{xx}-uv+{\rm i}\alpha u=f(x,t),\label{1.6} \end{equation}

(1.6)

\begin{equation} v_{t}+\beta v+\gamma(|u|^2)_{x}=g(x,t),\label{1.7} \end{equation}

(1.7)

其中复值函数$u(x,t)$ 表示高频电场的包络,实值函数 $v(x,t)$ 表示短波的振幅, 外力项函数 $g(x,t)$ 和 $f(x,t)$ 均表示依赖于时间的外力项. 长波-短波共振方程组来源于考虑重力和毛细模式的表面波及内部波(参见文献[4]). 在激光物理中,长波-短波共振方程组描述高频电子激光的碰撞效果及稠密低频离子 的混动行为(参见文献[35, 37]).

由于它的重要性,长波-短波共振方程组已经被广泛地研究,参见文献 [3, 5, 6, 21, 35, 52].特别地,文献 [52] 研究了格点长波-短波共振方程组(1.1)-(1.2),作者首先证明了方程组的解算子生成的过程存在紧致核截面,然后给出核截面的 Kolmogorov $\varepsilon$-entropy熵的上界估计,最后,证明了核截面的上半连续性.

本文的主要目的是估计文献 [52] 中所得的核截面的分形维数. 正如文献[48] 所指出,有限分形维数的重要性体现为: 对某个度量空间$E$,若 ${\cal A}$是它的紧子集,且${\rm dim}_f({\mathcal A}) < m/2$,其中$m$是某个自然数,则存在Lipschitz内射 $\varphi: {\mathcal A}\mapsto {\mathbb R}^m$,并且 ${\cal A}$ 的逆映射是Hölder连续的.这个性质表明 ${\cal A}$ 可以被放入到映 ${\mathbb R}^m$中的某个紧集到 ${\cal A}$的连续映射的图中.

不变集的分形维数已被广泛地研究^{[10, 18, 34, 39]}. 值得强调的是, 文献[18]证明了吸引子存在有限分形维数的一个准则,该准则是 Ladyzhenskaya的关于不变集的有限维数定理(见文献[34])的一个推广. 后来,文献 [48] 利用 文献[18] 中的思想方法来证明Hilbert空间中不变子集存在有限分形维数的准则.

在这篇文章中,我们将用文献 [48] 中法则去估计文献 [52] 中所得的核截面的分形维数. 本文的主要任务是证明方程组(1.1)-(1.2)所生成的过程: (1)在核截面 ${\cal K}(\tau)$上有Lipschitz性质; (2)在吸收集 ${\mathcal B}_0$中有压榨性质. 这里想指出的是,格点长波-短波共振方程组含有非线性项$(B(|u|^{2}))_{m}$,这给我们 在证明 Lipschitz 性质和压榨性质时带来了困难,因此需要做细致的计算和分析来处理这一项.

先介绍相关空间和算子. 记

\begin{equation} \ell^{2} = \Big \{u=(u_{m})_{m\in {\bf {\Bbb Z}}},u_{m}\in {\bf {\Bbb C}}:\sum\limits_{m\in{\bf \mathbb {Z}}}|u_{m}|^{2} < +\infty\Big\},\label{2.1} \end{equation}

(2.1)

\begin{equation} l^{2} = \Big \{v=(v_{m})_{m\in {\bf {\Bbb Z}}},v_{m}\in {\bf {\Bbb R}}:\sum\limits_{m\in{\bf {\Bbb Z}}}v_{m}^{2} < +\infty\Big\}.\label{2.2} \end{equation}

(2.2)

为了简洁起见,用 $X$ 表示 $\ell^2$ 或$l^2$,并在 $X$ 中定义内积与范数 $$ (u,v) = \sum\limits_{m\in{\bf {\Bbb Z}}}u_{m}\bar{v}_{m},\quad \|u\|^{2}=(u,u), \forall u=(u_{m})_{m\in {\bf {\Bbb Z}}}, \,v=(v_{m})_{m\in{\bf {\Bbb Z}}}\in X, $$ 其中 $\bar{v}_{m}$ 是 $v_m$ 的共轭. 同时,定义从 $X$ 到 $X$ 的线性算子 $B^*$ 如下 $$ (B ^{*}u)_m =u_{m-1}-u_{m},\quad \forall\,m\in {\mathbb Z}, \quad\forall\,u=(u_m)_{m\in{\mathbb Z}}\in X . $$ 可验证 $B^*$ 与 $B$ 互为伴随算子,且

\begin{eqnarray}\label{2.3} (Bu,v)=(u,B^*v),(Au,v)=(Bu,Bv), \forall u,\,v \in X. \end{eqnarray}

(2.3)

通过简单的计算有 $$ \|Bu\|^{2} =\sum\limits_{m\in{\bf {\Bbb Z}}}|u_{m+1}-u_{m}|^2 \leqslant 2\sum\limits_{m\in{\bf {\Bbb Z}}}(|u_{m+1}|^2+|u_{m}|^2) =4\|u\|^2 ,\forall u \in X. $$

为把格点方程组(1.1)-(1.3)写成向量形式,记 $$ \ell^{2} = (\ell^{2},(\cdot,\cdot),\|\cdot\|),\, l^{2}=(l^{2},(\cdot,\cdot),\|\cdot\|), $$ 则$\ell^{2}$和 $l^{2}$均为Hilbert空间. 记$ E=\ell^{2}\times l^{2} $,并在其中定义内积: 对于任意的$\psi^{(j)}=(u^{(j)},v^{(j)})^{T}\in E$,$j=1,2$,

\begin{equation} (\psi^{(1)},\psi^{(2)})_{E} = (u^{(1)},u^{(2)})+(v^{(1)},v^{(2)}) = \sum\limits_{m\in{\bf {\Bbb Z}}}(u_{m}^{(1)}\overline{u}_{m}^{(2)} + v_{m}^{(1)}v_{m}^{(2)}\big), \label{2.4} \end{equation}

(2.4)

$$ \|\psi\|_{E}^{2} = (\psi,\psi)_{E},\,\forall \psi\in E, $$ 其中$\overline{u}_m^{(2)}$ 表示 $u^{(2)}_m$ 的共轭.显然,$E$为Hilbert空间. 记 $$ u=(u_{m})_{m \in {\bf {\Bbb Z}}}, v=(v_{m})_{m \in {\bf {\Bbb Z}}},B|u|^{2}=(B(|u|^{2}))_{m \in {\bf {\Bbb Z}}}, f(t)=(f_{m}(t))_{m\in{\bf{\Bbb Z}}}, $$ $$ g(t)=(g_{m}(t))_{m \in {\bf {\Bbb Z}}}, u_{\tau}=(u_{m,\tau})_{m\in {\bf {\Bbb Z}}}, v_{\tau}=(v_{m,\tau})_{m \in {\bf {\Bbb Z}}}, $$ 则方程组(1.1)-(1.3)可以写成如下形式

\begin{equation} {\rm i}\dot{u}-Au-uv+{\rm i}\alpha u=f(t),t>\tau,\label{2.5} \end{equation}

(2.5)

\begin{equation} \dot{v}+\beta v+\gamma B|u|^2=g(t),t>\tau,\label{2.6}\\ \end{equation}

(2.6)

\begin{equation} u(\tau)=u_{\tau},v(\tau)=v_{\tau}, \tau\in{\bf {\Bbb R}},\label{2.7} \end{equation}

(2.7)

其中算子$A$和$B$分别由(1.4)式和(1.5)式所定义. 进一步,我们把 (2.5)-(2.7)式写成$E$中关于时间$t$的抽象的一阶常微分方程的初值问题

\begin{equation} \dot{\psi}+\Theta \psi=F(\psi,t),t> \tau,\label{2.8} \end{equation}

(2.8)

\begin{equation} \psi(\tau)=\psi_{\tau}=(u_{\tau},v_{\tau})^{T} ,\tau\in {\bf {\Bbb R}},\label{2.9} \end{equation}

(2.9)

其中$\psi=(u,v)^T,F(\psi,t) =(-{\rm i}uv-{\rm i}f(t),g(t)-\gamma B|u|^2)^T$,且

\begin{eqnarray}\label{2.10} \begin{array}{cc} \Theta = \left(\begin{array}{cc} \alpha I+{\rm i} A &~ 0 \\ 0 &~\beta I \\ \end{array}\right). \end{array} \end{eqnarray}

(2.10)

为讨论初值问题 (2.5)-(2.7) 解的适定性, 我们需要假设外力项函数满足的一定的性质. 记 ${\mathcal C}_{b}({\mathbb R},X)$ 为从 ${\mathbb R}$ 到 $X$ 的连续有界函数全体,则$\forall f(t)\in {\mathcal C}_{b}({\mathbb R},X)$,有 $\sup\limits_{t\in {\bf {\Bbb R}}}\sum\limits_{m\in {\bf {\Bbb Z}}}|f_{m}(t)|^{2} < +\infty.$ 记

\begin{eqnarray}\label{2.11} {\mathcal H}&=&\Big\{f(t)=( f_{m}(t))_{m \in {\bf {\Bbb Z}}}\in {\mathcal C}_{b}({\bf {\Bbb R}},X):\mbox{对每个}\,\tau\in {\bf {\Bbb R}},\,\forall \,\varepsilon> 0,\,\exists M(\varepsilon,\tau)\in {\bf {\Bbb N}},\nonumber\\ &&\quad \mbox{使得}\sum\limits_{|m|\geqslant M(\varepsilon,\tau)}|f_{m}(s)|^{2}\leqslant\varepsilon,\,\forall s\leqslant\tau\Big\}. \end{eqnarray}

(2.11)

本文,我们需要以下假设.

假设 (H) 设常数 $\alpha,\beta,\gamma$ 和 $f(t),g(t)$ 满足

\begin{eqnarray}\label{sss} \min\{\alpha,\beta\}>2\sqrt{\frac{(8\gamma^{2}+1)} {\min\{\frac{\alpha}{2},\frac{\beta}{2}\}}\Big(\frac{|||f|||^{2}}{\alpha} +\frac{2|||g|||^{2}}{\beta}+\frac{8\gamma^{4}|||f|||^{4}}{\alpha^{4}\beta}\Big)}. \end{eqnarray}

(2.12)

下面我们给出方程组(2.8)-(2.9)已有的一些结果.

引理 2.1^[52] 设 $f(t)=(f_{m}(t))_{m \in {\bf {\Bbb Z}}}\in {\mathcal C}_{b}({\bf {\Bbb R}},\ell^{2})$,$g(t)=(g_{m}(t))_{m \in {\bf {\Bbb Z}}}\in {\mathcal C}_{b}({\bf {\Bbb R}},l^{2})$. 则对任意的初值 $\psi_{\tau}=(u_{\tau},v_{\tau})^{T}\in E $,方程组(2.8)--(2.9) 有唯一解 $\psi(t)=(u(t),v(t))^{T}\in E$ 且 $\psi(t)\in {\mathcal C}([\tau,+\infty),E)\cap {\mathcal C}^{1}((\tau,+\infty),E)$. 另外,解映射

\begin{eqnarray}\label{2.14} U(t,\tau):\psi_{\tau} = (u_{\tau},v_{\tau})^{T}\in E \mapsto \psi(t) = (u(t),v(t))^{T}\in E,\forall t\geqslant\tau, \end{eqnarray}

(2.13)

在 $E $ 上 生成了一个连续的过程 $\{U(t,\tau)\}_{t\geqslant\tau}$.

引理 2.2^[52] 设引理2.1的条件成立. 则方程组 (2.8)-(2.9)相应于初值 $\psi_{t-s}=(u_{t-s},$ $v_{t-s})^{T}\in E$ 的解$\psi(t)=(u(t),v(t))^{T}=U(t,t-s){\psi_{t-s}}\in E $满足 $$ \|\psi(t)\|^{2}_{E} \leqslant C_{0}{\mathrm e}^{-\theta s}+\frac{r^{2}_{0}}{2\theta},\forall s>0, $$ 其中$C_{0},\theta=\min\{\frac{\alpha}{2},\frac{\beta}{2}\}$ 和 $r_{0}$ 均为不依赖于 $t$ 和 $s$ 的常数. 也就是说,过程$\{U(t,\tau)\}_{t\geqslant\tau}$ 在 $E$上存在 一致有界吸收集${\mathcal B}_0\subset E$,即对任意有界集${\mathcal B}\subset E$, 存在时间$s({\mathcal B})>0$,使得 $$ U(t,t-s){\mathcal B}\subseteq {\mathcal B}_0,\,\forall s\geqslant s({\mathcal B}), $$ 其中${\mathcal B}_0={\mathcal B}(0,R_{0})\subset E$是$E$中以0为中心, $R_{0}:= \frac{r_{0}}{\sqrt{\theta}}$为半径的闭球.

由引理2.2知存在时间$t_{0}:= t_{0}({\mathcal B}_0)$,使得

\begin{eqnarray}\label{2.15} U(t,t-s){\mathcal B}_0\subseteq {\mathcal B}_0, \,\forall s\geqslant t_{0}. \end{eqnarray}

(2.14)

引理 2.3^[52] 设$f(t)=(f_{m}(t))_{m \in {\bf {\Bbb Z}}}\in {\mathcal H}(X=\ell^{2}), g(t)=(g_{m}(t))_{m \in {\bf {\Bbb Z}}}\in {\mathcal H}(X=l^{2})$,则 $U(t ,t-s)\psi_{t-s}=\psi(t)=(\psi_{m}(t))_{m \in {\bf {\Bbb Z}}}\in E $为初值问题(2.8)--(2.9)的解,其中$\psi_{t-s}\in {\mathcal B}_0$. 则对任意的$\varepsilon>0$,存在时间$T(\varepsilon,{\mathcal B}_0)>0$, $M(\varepsilon,s,{\mathcal B}_0)\in {\bf {\Bbb Z_{+}}}$使得

\begin{eqnarray}\label{2.16} \sum\limits_{|m|>M(\varepsilon,s,{\mathcal B}_0)}|(U(t,t-s)\psi_{t-s})_{m}|^{2}_{E} = \sum\limits_{|m|>M(\varepsilon,s,{\mathcal B}_0)}|\psi_{m}(t)|^{2}_{E }\leqslant\varepsilon^{2}, s\geqslant T(\varepsilon,{\mathcal B}_0), \end{eqnarray}

(2.15)

其中$|\psi_{m}|^{2}_{E}=|u_m|^2 +v_m^2$.

引理 2.4^[52] 设引理2.3的条件成立. 则 $\left\{U(t,\tau)\right\}_{t\geqslant \tau}$ 存在一族紧致核截面 $\{{\mathcal K}(\tau)\}_{\tau\in {\Bbb R}}\subset E$,满足

(1) 紧致性: 对每个 $\tau\in {\Bbb R}$,${\mathcal K}(\tau)$ 是 $E$ 中紧集,且 ${\mathcal K}(\tau) =\bigcap\limits_{T>0} \overline{\bigcup\limits_{s>T}U(\tau,\tau-s){\mathcal B}_{0}} \subset {\mathcal B}_{0};$

(2) 不变性: $U(t,\tau){\mathcal K}(\tau)={\mathcal K}(t), \,\forall t\geqslant \tau,\,\tau\in {\mathbb R};$

(3) 拉回吸引性: 对 $E$ 中的任何有界集 ${\mathcal B}$,都有 $$\lim\limits_{s\rightarrow +\infty } {\mathrm dist}_{E}(U(\tau,\tau-s){\mathcal B},{\mathcal K}(\tau))=0,$$ 其中 ${\rm dist}_{E}(Y_1,Y_2)=\sup\limits_{x\in Y_1}\inf\limits_{y\in Y_2}\|x-y\|_{E}$.

本部分估计引理2.4所得核截面的分形维数.

定义 3.1 对任意的 $\tau\in {\mathbb R}$, 核截面${\mathcal K}(\tau)$的分形维数定义为

\begin{eqnarray}\label{3.1} {\rm dim}_F {\mathcal K}(\tau)=\limsup\limits_{\varepsilon \rightarrow 0}\frac{\ln {\mathcal N} ({\mathcal K}(\tau),\varepsilon)}{\ln (1/ \varepsilon)}, \end{eqnarray}

(3.1)

其中${\mathcal N} ({\mathcal K}(\tau),\varepsilon)$ 表示$E$中直径不超过$2\varepsilon$的 覆盖${\mathcal K}(\tau)$所需球的最少个数.

关于向量空间中不变集的分形维数的一般定义(参见文献[18, 34, 39]).

记

\begin{eqnarray}\label{3.2} E^{(N)} = \{\psi=(\psi_{m})_{m \in {\bf {\mathbb Z}}}\in E\,\big |\,\psi_{m}=(0,0)^T\,\,{\mbox{当}} \,\,|m|>N\mbox{时}\}, \end{eqnarray}

(3.2)

则$E^{(N)}$是$E$中的$3(2N+1)$维子空间. 定义映射$P_{N}: E\mapsto E^{(N)}\subset E$ 如下 \begin{eqnarray*} \quad (P_{N}\psi)_{m}=\left\{ \begin{array}{ll} \psi_{m},|m|\leqslant N; \\ 0,\quad |m|> N, \end{array} \right. \quad \psi=(\psi_m)_{m\in {\Bbb Z}}\in E. \end{eqnarray*}

下面的引理在估计核截面的分形维数时起关键作用.

引理3.1 若假设(H) 成立且 引理2.4的条件成立. 则对任意的 $\tau\in {\mathbb R}$,存在不依赖于$\tau\in {\mathbb R}$ 的正数$T^{*}$,$L(T^{*})$,$N^{*}$ 和 $\eta\in (0,1/2)$,使得

(I) 对任意的$\psi^{(1)}_{\tau},\psi^{(2)}_{\tau}\in {\mathcal K}(\tau)$,有

\begin{eqnarray}\label{3.3} ||U(T^{*}+\tau,\tau)\psi^{(1)}_{\tau}-U(T^{*}+\tau,\tau)\psi^{(2)}_{\tau}||_{E} \leqslant L(T^{*})||\psi^{(1)}_{\tau}-\psi^{(2)}_{\tau}||_{E}, \end{eqnarray}

(3.3)

也即$U(T^{*}+\tau,\tau)$是${\mathcal K}(\tau)$上的 Lipschitz 映射.

(II) 存在投影算子 $P_{2N^{*}}: E\mapsto E^{(2N^{*})}$ 使得对每个 $\tau\in {\mathbb R}, \psi^{(1)}_{\tau},\psi^{(2)}_{\tau}\in{\mathcal K}(\tau)\subseteq {\mathcal B}_{0}$,有

\begin{eqnarray}\label{3.4} \|(I-P_{2N^*})\big[U(T^{*}+\tau,\tau) \psi^{(1)}_{\tau}-U(T^{*}+\tau,\tau)\psi^{(2)}_{\tau}\big]\|_{E} \leqslant \eta\|\psi^{(1)}_{\tau}-\psi^{(2)}_{\tau}\|_{E}, \end{eqnarray}

(3.4)

其中 $I$ 是投影算子.

证 由文献 [52,引理3.1]知, 存在$E$中以2为直径覆盖${\mathcal K}(\tau)$ 的一致有界闭子集. 对任意的$\tau\in {\mathbb R}$,记 $$ \psi^{(1)}(t)=(u^{(1)}(t),v^{(1)}(t))^T=U(t,\tau)\psi^{(1)}_{\tau}, $$ $$ \psi^{(2)}(t)=(u^{(2)}(t),v^{(2)}(t))^T=U(t,\tau)\psi^{(2)}_{\tau},\forall t\geqslant\tau $$ 为初值问题(2.8)-(2.9)的两个解, 其中$\psi^{(1)}_{\tau},$ $\psi^{(2)}_{\tau}\in{\mathcal K}(\tau)\subseteq {\mathcal B}_{0}$. 对$t-\tau\geqslant t_{0},\psi^{(1)}(t),$ $ \psi^{(2)}(t) \in {\mathcal K}(t)\subset {\mathcal B}_{0}$. 文后记 \begin{eqnarray*} &\psi_{d}(t)=\psi^{(1)}(t)-\psi^{(2)}(t),\quad u_{d}(t)=u^{(1)}(t)-u^{(2)}(t),\quad & v_{d}(t)=v^{(1)}(t)-v^{(2)}(t). \end{eqnarray*} 由(2.8)-(2.9)式得

\begin{equation} \dot{\psi}_{d}+\Theta \psi_{d}=F(\psi^{(1)},t)-F(\psi^{(2)},t), ~ t>\tau,\label{3.6} \end{equation}

(3.5)

$$ \psi_{d}(\tau)=\psi^{(1)}_{\tau}-\psi^{(2)}_{\tau}. $$ 用$\psi_{d}$与(3.5)式在 $E$ 上作内积,然后取实部得

\begin{eqnarray}\label{3.7} \frac{1}{2}\frac{\mathrm{d}}{{\mathrm{d}}t}\|\psi_{d}(t)\|^{2}_{E} + \textbf{Re}\big(\Theta \psi_{d}(t),\psi_{d}(t)\big)_{E} =\textbf{Re}\big(F(\psi^{(1)},t)-F(\psi^{(2)},t),\psi_{d}(t)\big)_{E}, \forall t>\tau. \end{eqnarray}

(3.6)

由于 $\Theta: E\mapsto E$ 为有界线性算子(见(2.3)和(2.10)式), 而$F: E\times {\bf {\Bbb R}}\mapsto E$ 为满足局部 Lipschitz 条件的连续算子(参见文献[52,引理2.2]), 并且${\mathcal B}_0$为$E$中的有界集,因此有

\begin{equation} \textbf{Re}\big(\Theta \psi_{d},\psi_{d}\big)_{E} = \alpha (u_{d},u_{d})+\beta (v_{d},v_{d}),\label{SS} \end{equation}

(3.7)

\begin{eqnarray} &&\textbf{Re}\big(F(\psi^{1},t)-F(\psi^{2},t),\psi_{d}\big)_{E} \nonumber\\ &\leqslant& |\big(F(\psi^{1},t)-F(\psi^{2},t),\psi_{d}\big)_{E}| \leqslant ||F(\psi^{1},t)-F(\psi^{2},t)||_{E}||\psi_{d}||_{E}\nonumber\\ &\leqslant& \sqrt{\frac{2(16\gamma^{2}+2)}{\theta}}||\psi_{d}||^{2}_{E} %\nonumber\\ \leqslant 2\sqrt{\frac{8\gamma^{2}+1}{\theta}}\sqrt{\frac{1}{\alpha}|||f|||^{2} +\frac{2|||g|||^{2}}{\beta} +\frac{8\gamma^{4}|||f|||^{4}}{\alpha^{4}\beta}}||\psi_{d}||^{2}_{E}.\label{3.8} \end{eqnarray}

(3.8)

由(3.6),(3.7) 和 (3.8)式得

\begin{eqnarray}\label{3.9} \frac{\mathrm{d}}{{\mathrm{d}}t}\|\psi_{d}(t)\|^{2}_{E} \leqslant 2(K_{2}-K_{1})\|\psi_{d}(t)\|^{2}_{E}, \forall t-\tau >t_{0} , \end{eqnarray}

(3.9)

其中$K_{1}=\min\{\alpha,\beta\},K_{2} =2\sqrt{\frac{8\gamma^{2}+1}{\theta}(\frac{1}{\alpha}|||f|||^{2} +\frac{2|||g|||^{2}}{\beta}+\frac{8\gamma^{4}|||f|||^{4}}{\alpha^{4}\beta})}$, 因此,

\begin{eqnarray}\label{3.10} \|\psi^{(1)}(T^*+\tau)-\psi^{(2)}(T^*+\tau)\| _{E} &=& \|\psi_{d}(T^*+\tau)\|_{E} \nonumber\\ &\leqslant& {\mathrm e}^{(K_{2}-K_{1})T^*}\|\psi^{(1)}(\tau)-\psi^{(2)}(\tau)\|_{E}, \end{eqnarray}

(3.10)

其中$T^*>0$ (将在文后(3.32)式中给出)是不依赖于$\tau$的常数. 由(3.10)式知$U(T^*+\tau,\tau)$是${\mathcal K}(\tau)$中的 Lipschitz 映射, 且Lipschitz常数为$L(T^*)={\mathrm e}^{(K_{2}-K_{1})T^*}$.

(II) 定义一个光滑函数 $\chi(x)\in {\mathcal C}^{1}({\bf {\Bbb R}}_+,[0, 1])$,满足 \begin{eqnarray*} \quad \chi(x)=\left\{ \begin{array}{ll} 0,0\leqslant x\leqslant 1; \\ 1,x\geqslant 2, \end{array} \right.\,{\mbox{且}} |\chi'(x)|\leqslant \chi_0 \,(\mbox{正常数}), \forall\,x\in {\bf {\Bbb R}}_+. \end{eqnarray*} 记 $$ p_d=(p_{dm})_{m\in {\Bbb Z}},q_d=(q_{dm})_{m\in {\Bbb Z}}, z_{d}=(z_{dm})_{m\in {\Bbb Z}}, $$ $$ p_{dm}=\chi(\frac{|m|}{M})u_{dm}, q_{dm}=\chi(\frac{|m|}{M})v_{dm}, z_{dm}=(p_{dm},q_{dm}), $$ 其中$M$为某个正数. 由于$(u^{(1)},v^{(1)})$ 和 $(u^{(2)},v^{(2)})$满足 (2.5)式,因此

\begin{eqnarray}\label{3.11} {\rm i}\dot{u}_{d}-Au_{d}+{\rm i}\alpha u_{d}=u^{(1)}v^{(1)}-u^{(2)}v^{(2)}. \end{eqnarray}

(3.11)

用$p_{d}$在$\ell^{2}$上与(3.11)式作内积,然后取虚部,得

\begin{eqnarray} && \frac{1}{2}\frac{\mathrm{d}}{{\mathrm{d}}t} \sum\limits_{m\in{\bf{\Bbb Z}}}\chi(\frac{|m|}{M})|u_{dm}|^{2} -\textbf{Im}(Au_d,p_d) + \alpha\sum\limits_{m\in{\bf{\Bbb Z}}}\chi(\frac{|m|}{M})|u_{dm}|^{2} \nonumber\\ &=&\textbf{Im}(u^{(1)}v^{(1)}-u^{(2)}v^{(2)},p_{d}).\quad \label{3.12} \end{eqnarray}

(3.12)

通过计算,我们有

\begin{eqnarray}\label{3.13} -\textbf{Im}(Au_d,p_d) & = & -\textbf{Im}(Bu_d,Bp_d) \nonumber\\ &=& -\textbf{Im}\sum\limits_{m\in{\bf{\Bbb Z}}}(u_{dm+1}-u_{dm}) \Big(\chi(\frac{|m+1|}{M}){\bar u_{dm+1}} -\chi(\frac{|m|}{M}){\bar u_{dm}}\Big)\nonumber\\ &=& \textbf{Im}\sum\limits_{m\in{\bf{\Bbb Z}}} \Big(\chi(\frac{|m+1|}{M}){\bar u_{dm+1}}u_{dm} +\chi(\frac{|m|}{M}){\bar u_{dm}}u_{dm+1}\Big)\nonumber\\ &=& \textbf{Im}\sum\limits_{m\in{\bf{\Bbb Z}}} \Big(\chi(\frac{|m+1|}{M}){\bar u_{dm+1}}u_{dm} -\chi(\frac{|m|}{M}){\bar u_{dm+1}}u_{dm}\Big)\nonumber\\ &\geqslant& -\sum\limits_{m\in{\bf{\Bbb Z}}} |\chi'(\frac{{\tilde m}}{M})|\frac{1}{M}|u_{dm+1}||u_{dm}| \nonumber\\ &\geqslant& -\frac{\chi_{0}}{M}\|\psi_{d}\|^2_{E}, \forall t >\tau, \end{eqnarray}

(3.13)

其中$\tilde{m}$ 为介于 $|m+1|$ 和 $|m|$ 中的某个常数. 由引理2.3知,存在 $t_1=t_{1}(\alpha,\beta,{\mathcal B}_0)>0,$ $ M_1=M_{1}(\alpha,\beta,t-\tau,{\mathcal B}_0)\in {\Bbb N}$使得

\begin{eqnarray} && \textbf{Im}\sum_{m\in {\Bbb Z}}\big(u^{(1)}_{m}v^{(1)}_{m}-u^{(2)}_{m}v^{(2)}_{m}\big) \chi(\frac{|m|}{M})( \bar {u} ^{(1)}_{m}-\bar{u}^{(2)}_{m})\nonumber\\ &\leqslant& \sum_{m\in {\Bbb Z}}\chi(\frac{|m|}{M}) |u^{(1)}_{m}v^{(1)}_{m}-u^{(2)}_{m}v^{(2)}_{m}| \,|u^{(1)}_{m}-u^{(2)}_{m}|\nonumber\\ &=& \sum_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u^{(1)}_{m}(v^{(1)}_{m}-v^{(2)}_{m}) + v^{(2)}_{m}(u^{(1)}_{m}-u^{(2)}_{m})| \,|u^{(1)}_{m}-u^{(2)}_{m}|\nonumber\\ &\leqslant& \frac{\alpha}{4}\sum_{m\in {\Bbb Z}} \chi(\frac{|m|}{M})|u^{(1)}_{m}-u^{(2)}_{m}|^{2} + \frac{\sqrt{\alpha\beta}}{2}\sum_{m\in {\Bbb Z}} \chi(\frac{|m|}{M}) |u^{(1)}_{m}-u^{(2)}_{m}||v^{(1)}_{m}-v^{(2)}_{m}|\nonumber\\ &\leqslant& \frac{\alpha}{2}\sum_{m\in {\Bbb Z}} \chi(\frac{|m|}{M})|u _{dm}|^{2} + \frac{\beta}{4}\sum_{m\in {\Bbb Z}} \chi(\frac{|m|}{M})|v_{dm}|^{2},\forall t-\tau> t_{1},\label{3.14} \forall M>M_{1}, \end{eqnarray}

(3.14)

故对任意的$t-\tau> t_{1},M> M_{1}$,有

\begin{eqnarray} \textbf{Im}(u^{(1)} v^{(1)}-u^{(2)}v^{(2)},p_{d}) \leqslant \frac{\alpha}{2}\sum_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u _{dm}|^{2} + \frac{\beta}{4}\sum_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|v_{dm}|^{2}.\label{3.15} \end{eqnarray}

(3.15)

由(3.12)-(3.15) 式知,

\begin{eqnarray}\label{3.16} &&\frac{1}{2}\frac{\mathrm{d}}{{\mathrm{d}}t} \sum\limits_{m\in{\bf{\Bbb Z}}}\chi(\frac{|m|}{M})|u_{dm}|^{2} + \frac{\alpha}{2}\sum\limits_{m\in{\bf{\Bbb Z}}} \chi(\frac{|m|}{M})|u_{dm}|^{2}\nonumber\\ & \leqslant& \frac{\beta}{4}\sum\limits_{m\in{\bf{\Bbb Z}}} \chi(\frac{|m|}{M})v_{dm}^{2} + \frac{\chi_{0}}{M}\|\psi_{d}\|^{2}_{E},\forall t-\tau> t_{1}, M>M_{1}. \end{eqnarray}

(3.16)

类似(3.11)式,$v_d$ 满足

\begin{eqnarray}\label{3.17} \dot{v}_{d}+\beta v_{d} +\gamma B(|u^{(1)}|^{2}-|u^{(2)}|^{2})=0. \end{eqnarray}

(3.17)

用$q_{d}$在$l^{2}$上与(3.17)式作内积,得

\begin{eqnarray}\label{3.18} \frac{1}{2}\frac{\mathrm{d}}{{\mathrm{d}}t} \sum\limits_{m\in{\bf {\Bbb Z}}} \chi(\frac{|m|}{M})v_{dm}^{2} + \beta\sum\limits_{m\in{\bf {\Bbb Z}}} \chi(\frac{|m|}{M})v_{dm}^{2} +\gamma\big(B(|u^{(1)}|^{2}-|u^{(2)}|^{2}),q_{d}\big)=0. \end{eqnarray}

(3.18)

下面估计$\gamma\big(B(|u^{(1)}|^{2}-|u^{(2)}|^{2}),q_{d}\big)$. 首先,

\begin{eqnarray}\label{3.21} \big|\gamma\big(B(|u^{(1)}|^{2}-|u^{(2)}|^{2}),q_{d}\big)\big| & =& \bigg|\gamma\sum\limits_{m\in{\bf {\Bbb Z}}} \chi(\frac{|m|}{M}) \big(B(|u^{(1)}|^{2}-|u^{(2)}|^{2})\big)_{m}v_{dm}\bigg|\nonumber\\ &=& \bigg|\gamma\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M}) \Big[(|u^{(1)}_{m+1}|^2-|u^{(2)}_{m+1}|^2) -(|u^{(1)}_{m}|^2-|u^{(2)}_{m}|^2)\Big]v_{dm}\bigg|\nonumber\\ &\leqslant& I_1+I_2, \end{eqnarray}

(3.19)

其中 $$ I_1=\gamma\bigg|\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})(|u^{(1)}_{m}|^2 -|u^{(2)}_{m}|^2)v_{dm}\bigg|, $$ $$ I_2=\gamma\bigg|\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})(|u^{(1)}_{m+1}|^2 -|u^{(2)}_{m+1}|^2)v_{dm}\bigg|. $$ 由引理2.3和柯西不等式知,存在$t_2=t_{2}(\alpha,\beta,\gamma,{\mathcal B}_0)>0,$ $M_2=M_{2}(\alpha,\beta,\gamma,t-\tau,{\mathcal B}_0)\in {\Bbb N}$,使得

\begin{eqnarray}\label{3.22} I_1 &\leqslant& \frac{\beta}{4}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})v^2_{dm} +\frac{\gamma^2}{\beta}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})\Big[(|u^{(1)}_{m}|^2-|u^{(2)} _{m}|^2)\Big]^2 \nonumber\\ &\leqslant& \frac{\beta}{4}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})v^2_{dm} +\frac{\gamma^2}{\beta}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M}) |u^{(1)}_{m}-u^{(2)}_{m}|^2(|u^{(1)}_{m}|^2+|u^{(2)}_{m}|^2) \nonumber\\ &\leqslant& \frac{\beta}{4}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})v^2_{dm} +\frac{\alpha}{4}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})|u_{dm}|^2, t-\tau> t_{2},M>M_{2}. \end{eqnarray}

(3.20)

其次,由柯西不等式得

\begin{eqnarray}\label{3.23} I_2 &\leqslant& \frac{\beta}{4}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})v^2_{dm} +\frac{\gamma^2}{\beta}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})\Big[(|u^{(1)}_{m+1}|^2 -|u^{(2)}_{m+1}|^2)\Big]^2 \nonumber\\ &\leqslant& \frac{\beta}{4}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})v^2_{dm} + \frac{2\gamma^2 \|\psi_{d}\|_E^2}{\beta} \sum\limits_{m\in{\bf {\Bbb Z}}} \chi(\frac{|m|}{M}) (|u^{(1)}_{m+1}|^2+|u^{(2)}_{m+1}|^2). \end{eqnarray}

(3.21)

再由文献[52,(3.15)式]知,

\begin{eqnarray}\label{3.24} \sum\limits_{m\in {\bf {\Bbb Z}}}\chi(\frac{|m|}{M})|u^{(j)}_{m+1}(t)|^2 &\leqslant& \frac{1}{\alpha}\int_{\tau}^t \Big(\sum\limits_{|m|\geqslant M}|f_{m+1}(\sigma)|^2 \Big){\mathrm e}^{-\alpha(t-\sigma)}{\mathrm d}\sigma \nonumber\\ && +R^2_0{\mathrm e}^{-\alpha(t-\tau)} +\frac {2\chi_0R^2_0}{\alpha M}, \quad j=1,2, \forall\,t-\tau\geqslant t_{0}, \end{eqnarray}

(3.22)

其中$t_0$由(2.14)式确定. 由(3.16),(3.18)式和(3.19)-(3.22) 式得

\begin{eqnarray}\label{3.25} && \frac{\mathrm{d}}{{\mathrm{d}}t} \sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})|\psi_{dm}|^2_{E} +\vartheta\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})|\psi_{dm}|^2_{E} \leqslant I_3\|\psi_{d}\|^{2}_{E}, \end{eqnarray}

(3.23)

其中$\displaystyle\vartheta=\min\{\alpha/2,\beta/2\}$, $t-\tau>\max\{t_0,t_1,t_2\}$,$M>\max\{M_1,M_2\}$,

\begin{eqnarray}\label{3.26} I_3 &=&\frac{8\gamma^2}{\beta}\bigg[\frac{1}{\alpha}\int_{\tau}^t \Big(\sum\limits_{|m|\geqslant M}|f_{m+1}(\sigma)|^2 \Big){\mathrm e}^{-\alpha(t-\sigma)}{\mathrm d}\sigma +R^2_0{\mathrm e}^{-\alpha(t-\tau)} +\frac {2\chi_0R^2_0}{\alpha M}\bigg] +\frac {2\chi_0}{M} \nonumber\\ &\leqslant& \frac{8\gamma^2}{\alpha^2\beta}\sup\limits_{t\in {\Bbb R}} \sum\limits_{|m|\geqslant M}|f_{m+1}(t)|^2 +\Big(\frac{8\gamma^2}{\beta}\frac {2\chi_0R^2_0}{\alpha M} +\frac {2\chi_0}{M}\Big) +\frac{8\gamma^2}{\beta}R^2_0{\mathrm e}^{-\alpha(t-\tau)}. \end{eqnarray}

(3.24)

对(3.23)式从$\tau$ 到 $t$应用格朗沃尔不等式,有

\begin{eqnarray} \sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})|\psi_{dm}(t)|^{2}_{E} \leqslant {\mathrm e}^{-\vartheta(t-\tau)}\sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})|\psi_{dm}(\tau)|^{2}_{E} + \int_{\tau}^{ t}I_3\|\psi_{d}(y)\|^{2}_{E} {\mathrm e}^{-\vartheta(t-y)}{\mathrm{d}}y,\quad\label{3.27} \end{eqnarray}

(3.25)

由(3.10)式得,

\begin{eqnarray}\label{3.28} \int_{\tau}^{ t}I_3\|\psi_{d}(y)\|^{2}_{E} {\mathrm e}^{-\vartheta(t-y)}{\mathrm{d}}y &\leqslant& \|\psi_{d}(\tau)\|^{2}_{E}\int_{\tau}^{ t} I_3{\mathrm e}^{-\vartheta(t-y) +2(K_{2}-K_{1})(y-\tau)}{\mathrm{d}}y. \end{eqnarray}

(3.26)

注意到

\begin{eqnarray} && \int_{\tau}^{ t} \Big (\frac{2\gamma^2}{\beta}\frac {2\chi_0R^2_0}{\alpha M} +\frac {2\chi_0}{M}\Big){\mathrm e}^{-\vartheta(t-y) +2(K_{2}-K_{1})(y-\tau)}{\mathrm{d}}y \nonumber\\ &\leqslant& \frac 1M\Big(\frac{8\gamma^2}{\beta}\frac {2\chi_0R^2_0}{\alpha } +2\chi_0\Big)\frac{\chi_{0}{\mathrm e}^{2(K_{2}-K_{1})(t-\tau)}}{ (\vartheta+2K_{2}-2K_{1})},\label{3.29} \end{eqnarray}

(3.27)

\begin{eqnarray} && \frac{8\gamma^2}{\beta} \int_{\tau}^{ t} R^2_0{\mathrm e}^{-\alpha(y-\tau)}{\mathrm e}^{-\vartheta(t-y) +2(K_{2}-K_{1})(y-\tau)}{\mathrm{d}}y \nonumber\\ &\leqslant& \frac{8\gamma^2R^2_0}{\beta[{2(K_{2}-K_{1})+\vartheta-\alpha}]} {\mathrm e}^{(2K_{2}-2K_{1}-\alpha)(t-\tau)},\label{3.30} \end{eqnarray}

(3.28)

\begin{eqnarray} && \int_{\tau}^{ t}\frac{8\gamma^2}{\alpha^2\beta}\sup\limits_{y\in {\Bbb R}} \sum\limits_{|m|\geqslant M}|f_{m+1}(y)|^2{\mathrm e}^{-\vartheta(t-y) +2(K_{2}-K_{1})(y-\tau)}{\mathrm{d}}y\nonumber\\ &\leqslant& \frac{8\gamma^{2}|||f|||^2}{\alpha^{2}\beta(\vartheta+2K_{2}-2K_{1})} {\mathrm e}^{(2K_{2}-2K_{1})(t-\tau)}.\label{3.31} \end{eqnarray}

(3.29)

对任意的$M> \max\{M_1,M_2\}$, $t-\tau> \max\{t_0,t_1,t_2\}$,由(3.24)--(3.29) 式得

\begin{eqnarray} \sum\limits_{m\in{\bf {\Bbb Z}}}\chi(\frac{|m|}{M})|\psi_{dm}(t)|^{2}_{E} &\leqslant& \bigg[{\mathrm e}^{-\vartheta(t-\tau)} +\frac{8\gamma^2R^2_0}{[2(K_{2}-K_{1})+ \vartheta-\alpha]\beta}{\mathrm e}^{(2K_{2}-2K_{1}-\alpha)(t-\tau)}\nonumber\\ && +\frac{8\gamma^2}{\alpha^2\beta(\vartheta+2K_{2}-2K_{1})} {\mathrm e}^{(2K_{2}-2K_{1})(t-\tau)}\bigg] \|\psi_{d}(\tau)\|^{2}_{E}\nonumber\\ && +\frac{1}{M}\Big(2\chi_{0}+\frac{16\chi_{0}\gamma^2R^2_0} {\alpha\beta}\Big)\frac{\chi_{0}{\mathrm e}^{2(K_{2}+K_{1})(t-\tau)}} {\vartheta+2K_{2}-2K_{1}}\|\psi_{d}(\tau)\|^{2}_{E}. \quad\label{3.32} \end{eqnarray}

(3.30)

由(2.12)式知$K_{1}>K_{2}$. 取定$T^*\geqslant \max\{t_0,t_1,t_2\}$, 使得

\begin{equation} {\mathrm e}^{-\vartheta T^*} < \frac {1}{16},\quad \frac{8\gamma^2R^2_0}{\beta[2(K_{2}-K_{1})+ \vartheta-\alpha]}{\mathrm e}^{(2K_{2}-2K_{1}-\alpha)T^*} < \frac {1}{16},\label{S1} \end{equation}

(3.31)

\begin{equation} \frac{8\gamma^2}{\alpha^2\beta(\vartheta+2K_{2}-2K_{1})}{\mathrm e}^{(2K_{2}-2K_{1})T^*} < \frac {1}{16}.\label{S2} \end{equation}

(3.32)

对上述确定的$T^*$,取$N^*\geqslant \max\{M_1,M_2\}$,使得

\begin{eqnarray}\label{3.33} \frac{1}{M}\Big(2\chi_{0}+\frac{16\chi_{0}\gamma^2R^2_0} {\alpha\beta}\Big)\frac{\chi_{0}{\mathrm e}^{2(K_{2}-K_{1})T^*}} {\vartheta+2K_{2}-2K_{1}} < \frac {1}{16},\forall\,M\geqslant N^*. \end{eqnarray}

(3.33)

因此,由(3.30)式得

\begin{eqnarray}\label{3.34} \sum\limits_{|m|>2N^*}|\psi_{dm}(T^{*}+\tau)|^{2}_{E} \leqslant \sum\limits_{m\in{\bf{\Bbb Z}}} \chi(\frac{|m|}{N^*})|\psi_{dm}(T^{*}+\tau )|^{2}_{E} \leqslant \eta^{2}\|\psi_{d}(\tau)\|^{2}_{E}, \end{eqnarray}

(3.34)

其中

\begin{eqnarray} \eta^2 &=&{\mathrm e}^{-\vartheta T^*} +\frac{8\gamma^2R^2_0}{\beta[2(K_{2}-K_{1}) + \vartheta-\alpha]}{\mathrm e}^{(2K_{2}-2K_{1}-\alpha)T^*} \nonumber\\ &&+ \frac{8\gamma^2}{\alpha^2\beta(\vartheta+2K_{2}-2K_{1})} {\mathrm e}^{(2K_{2}-2K_{1})T^*} \nonumber\\ && +\frac{1}{M}\Big(2\chi_{0}+\frac{16\chi_{0}\gamma^2R^2_0} {\alpha\beta}\Big)\frac{\chi_{0}{\mathrm e}^{2(K_{2}-K_{1})T^*}} {\vartheta+2K_{2}-2K_{1}} < \frac{1}{4}. \end{eqnarray}

(3.35)

由(3.34)式得

\begin{eqnarray}\label{3.35} \|(I-P_{2N^*}) \big[U(T^{*}+\tau,\tau)\psi^{(1)}_{\tau} -U(T^{*}+\tau,\tau)\psi^{(2)}_{\tau}\big]\|_{E} \leqslant \eta\|\psi^{(1)}_{\tau}-\psi^{(2)}_{\tau}\|_{E}. \end{eqnarray}

(3.36)

引理3.1证明完毕.

定理 3.1 若假设(H) 成立且 引理2.4的条件成立, $\{{\cal K}(\tau)\}_{\tau\in {\Bbb R}}$为过程$\{U(t,\tau)\}_{t\geqslant \tau}$ 的核截面. 则对任意的$\tau\in {\Bbb R}$,核截面${\mathcal K}(\tau)$的分形维数满足

\begin{eqnarray}\label{3.36} \dim_F{\mathcal K}(\tau) \leqslant 3(4N^*+1) \cdot \ln\Big(1+\frac{8(1+{\mathrm e}^{(K_1+K_2)T^*})}{\frac{1}{2}-\eta}\Big) \cdot \Big (\ln\frac{2}{\frac{3}{2}+\eta}\Big)^{-1}, \end{eqnarray}

(3.37)

其中$T^*$,$N^*$,$\eta$,$K_1$ 和 $K_2$ 为不依赖于$\tau$的常数.

证 由文献[48,引理3.1和定理2.1]可得该定理的结果. 证明完毕.