## 脉冲离散Ginzburg-Landau方程组的统计解及其极限行为

1 温州大学数理学院 浙江温州 325035

2 塞维利亚大学数学系 西班牙塞维利亚 41012

## Statistical Solutions and Its Limiting Behavior for the Impulsive Discrete Ginzburg-Landau Equations

Zhao Caidi,1, Jiang Huite1, Li Chunqiu1, Tomás Caraballo2

1 Department of Mathematics, Wenzhou University, Zhejiang Wenzhou 325035

2 Departmento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Mathmáticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain

 基金资助: 国家自然科学基金.  11971356浙江省自然科学基金.  LY17A010011

 Fund supported: the NSFC.  11971356the NSF of Zhejiang Province.  LY17A010011

Abstract

In this article we first prove the global well-posedness of the impulsive discrete Ginzburg-Landau equations. Then we establish that the generated process by the solution operators possesses a pullback attractor and a family of invariant Borel probability measures. Further, we formulate the definition of statistical solution for the addressed impulsive system and prove the existence. Our results reveal that the statistical solution of the impulsive system satisfies merely the Liouville type theorem piecewise, which implies that the Liouville type equation for impulsive system will not always hold true on the interval containing any impulsive point. Finally, we prove that the statistical solution of the impulsive discrete Ginzburg-Landau equations converges to that of the impulsive discrete Schrödinger equations.

Keywords： Statistical solution ; Impulsive differential equation ; Liouville type theorem ; Discrete complex Ginzburg-Landau equation ; Discrete Schrödinger equation

Zhao Caidi, Jiang Huite, Li Chunqiu, Tomás Caraballo. Statistical Solutions and Its Limiting Behavior for the Impulsive Discrete Ginzburg-Landau Equations. Acta Mathematica Scientia[J], 2022, 42(3): 784-806 doi:

## 1 引言

$$${\rm i}\frac{{\mathrm d}u_m}{{\mathrm d}t} -(\alpha-{\rm i}\epsilon)(2u_{m}-u_{m+1}-u_{m-1}) +{\rm i}\kappa u_{m} +\beta|u_m|^{2\gamma}u_m=g_m(t), \; t>\tau, \, t\neq \tau_j, \, \, m, j\in {\Bbb Z},$$$

$$$u_m(\tau_j^+)-u_m(\tau_j)=\phi_{mj}(u_m(\tau_j)), \; m, j\in {\mathbb Z}, \; \tau_j\in {\mathbb R},$$$

$$$u_m(\tau^+)=\lim\limits_{s\rightarrow \tau^+}u_m(s)=u_{m, \tau^+}, \; m\in {\mathbb Z},$$$

$\begin{eqnarray} (Au, v)=(B^*Bu, v)=(Bu, Bv), \; \forall\, u, v\in \ell^2, \end{eqnarray}$

$\begin{eqnarray} \left\{ \begin{array}{ll} \|Bu\|^2=\|B^*u\|^2\leqslant 4\|u\|^2, \;& \forall\, u\in \ell^2, \\ \|Au\|^2\leqslant 16\|u\|^2, \; &\forall\, u\in \ell^2. \end{array} \right. \end{eqnarray}$

(H1) 对任意的$m, j\in {\Bbb Z}$, $\phi_{mj}(0)=0$, 存在一个常数$L>0$使得

$$$|\phi_{mj}(z')-\phi_{mj}(z'')| \leqslant L|z'-z''|, \; \forall\, z', z''\in {\Bbb C},$$$

$$$\sigma:=\kappa-\frac{1}{\eta}\ln(2+2L^2)>0,$$$

(H2) $g(\cdot)\in C(\ell^2)$, 且对任意的$t\in {{\Bbb R}} $$\int_{-\infty}^t {\rm e}^{\sigma s}\|g(s)\|^2{\mathrm d}s<+\infty . 引理2.1 假设 \alpha , \epsilon , \kappa , \beta 都是正的常数, 1<\gamma<+\infty$$ \rm ({H1})–({H2})$成立.则对于每个初始时刻$\tau$和初始值$u_{\tau^+}\in\ell^2$, 问题(2.3)–(2.5)存在唯一的解$u$满足

将方程(2.3)改写成下面形式

$\begin{eqnarray} \frac{{\mathrm d} u}{{\mathrm d}t} = F^\epsilon(u, t) := (-\epsilon-{\rm i}\alpha) Au-\kappa u+{\rm i}\beta f(|u|^{2})u-{\rm i}g(t), \; t>\tau, \, t\neq \tau_j, \, j\in {\Bbb Z}, \end{eqnarray}$

$$$f(x)=x^\gamma, \; x\in {{\Bbb R}} _+.$$$

$\begin{eqnarray} \|\tilde{f}(u)-\tilde{f}(v)\|^2 \leqslant M_f({\mathcal B})\|u-v\|^2, \; \forall\, u, v\in {\mathcal B}, \end{eqnarray}$

$\begin{eqnarray} \left\{ \begin{array}{ll} \frac{{\mathrm d}y(t)}{{\mathrm d}t}+ay(t) \leqslant q(t), &\; t\neq \tau_j, \, j\in {\Bbb Z}, \\ y(\tau_j^+)-y(\tau_j) \leqslant by(\tau_j), &\; j\in {\Bbb Z}, \\ y(s^+) \leqslant y_{0}, &\; s\in {{\Bbb R}} , \end{array} \right. \end{eqnarray}$

$\begin{eqnarray} \|u(t)\|^2 \leqslant \|u_{\tau^+}\|^2{\rm e}^{-\sigma(t-\tau)} +\frac{{\rm e}^{-\sigma t}}{\kappa} \int_\tau^t{\rm e}^{\sigma \theta}\|g(\theta)\|^2{\mathrm d}\theta, \; \tau<t\leqslant T_*, \end{eqnarray}$

记$u(\cdot)=u(\cdot; \tau, u_{\tau^+})$为问题(2.3)–(2.5)在初始时刻$\tau $$u_{\tau^+} 为初值的解.用 u(\cdot) 与(2.9)式在 \ell^2 上作内积并取实部, 可得 \begin{eqnarray} \frac{{\mathrm d}}{{\mathrm d}t}\|u(t)\|^2 +\kappa \|u(t)\|^2 \leqslant \frac{\|g(t)\|^2}{\kappa}, \; t\neq\tau_j, j\in {\Bbb Z}. \end{eqnarray} 由假设(H1)得 \begin{eqnarray} \|u(\tau_j^+)\|^2 &=&\sum\limits_{m\in {\Bbb Z}}|u_m(\tau^+_j)|^2 =\sum\limits_{m\in {\Bbb Z}}\big|u_m(\tau_j)+\phi_{mj}(u_m(\tau_j))\big|^2 \\ &\leqslant& 2\sum\limits_{m\in {\Bbb Z}}|u_m(\tau_j)|^2 +2\sum\limits_{m\in {\Bbb Z}}|\phi_{mj}(u_m(\tau_j))|^2 \\ &\leqslant& (2+2L^2)\sum\limits_{m\in {\Bbb Z}}|u_m(\tau_j)|^2 = (2+2L^2)\|u(\tau_j)\|^2. \end{eqnarray} y(t)=\|u(t)\|^2 , 对(2.15)–(2.16)式应用引理2.2可得 $$\|u(t)\|^2 \leqslant \|u_{\tau^+}\|^2(2+2L^2)^{n(\tau, t)}{\rm e}^{-\kappa(t-\tau)} +\frac{1}{\kappa}\int_\tau^t(2+2L^2)^{n[s, t)} {\rm e}^{-\kappa(t-s)}\|g(s)\|^2{\mathrm d}s, \; \, \forall\, t>\tau.$$ 而(1.4)式意味着 因此, 由(2.8)式可得 \begin{eqnarray} (2+2L^2)^{n(\tau, t)}{\rm e}^{-\kappa(t-\tau)} \leqslant {\rm e}^{-\sigma(t-\tau)} \; \mbox{和}\; (2+2L^2)^{n[s, t)} {\rm e}^{-\kappa(t-s)} \leqslant {\rm e}^{-\sigma(t-s)}. \end{eqnarray} 将(2.18)式代入(2.17)式得到(2.14)式.证明完毕. 结合引理2.1和引理2.3知问题(2.3)–(2.5)存在唯一的全局解. 定理2.1 假设 \alpha , \epsilon , \kappa , \beta 都是正的常数, 1<\gamma<+\infty$$ \rm ({H1})–({H2})$成立, 则对于每一个给定的初始时刻$\tau\in {{\Bbb R}}$和初值$u(s^+)\in \ell^2$, 问题(2.3)–(2.5)存在唯一的解$u \in PC([\tau, +\infty);\ell^2)\cap PC^1((\tau, +\infty);\ell^2)$满足

$\begin{eqnarray} \|u(t)\|^2 \leqslant \|u_{\tau^+}\|^2{\rm e}^{-\sigma(t-\tau)} +\frac{{\rm e}^{-\sigma t}}{\kappa} \int_\tau^t{\rm e}^{\sigma \theta}\|g(\theta)\|^2{\mathrm d}\theta, \; \forall\, t>\tau. \end{eqnarray}$

$$$\|u^{(1)}(t)-u^{(2)}(t)\|^2 \leqslant \|u^{(1)}_{\tau^+}-u^{(2)}_{\tau^+}\|^2 {\rm e}^{-(\sigma+\kappa-2\ell(R)) (t-\tau)}, \; \forall\, t>\tau,$$$

$$${\rm i}\frac{{\mathrm d} v}{{\mathrm d}t} -(\alpha-{\rm i}\epsilon) Av+{\rm i}\kappa v+\beta(\tilde{f}(u^{(1)})-\tilde{f}(u^{(2)}))=0, \; t>\tau, \, t\neq \tau_j, \, j\in {\Bbb Z},$$$

$$$v(\tau_j^+)-v(\tau_j)=\phi_j(u^{(1)}(\tau_j))-\phi_j(u^{(2)}(\tau_j)), \; \, j\in {\Bbb Z},$$$

$$$v(\tau^+)=u^{(1)}_{\tau^+}-u^{(2)}_{\tau^+}, \; \tau\in {{\Bbb R}} .$$$

$-{\rm i}v$与(2.21)式两边在$\ell^2$上作内积并取实部, 得

$\begin{eqnarray} \frac12\frac{{\mathrm d}}{{\mathrm d}t}\|v\|^2 +\epsilon \|B v\|^2+\kappa \|v\|^2+{\bf{Im}}\beta(\tilde{f}(u^{(1)})-\tilde{f}(u^{(2)}), v) =0, \; t\neq\tau_j, j\in {\Bbb Z}. \end{eqnarray}$

$$$\big|{\bf{Im}} (\tilde{f}(u^{(1)})-\tilde{f}(u^{(2)}), v)\big| \leqslant \ell(R)\|v\|^2,$$$

$\begin{eqnarray} {\mathcal D}_\sigma =\Big\{\widehat{D} =\{D(t):t\in{{\Bbb R}} \}\subseteq{\cal P}(\ell^2) |\lim\limits_{\tau\rightarrow -\infty} ({\rm e}^{\sigma \tau} \sup\limits_ {u\in D(\tau)}\|u(\cdot)\|^2)=0\Big\}. \end{eqnarray}$

(1) 设$\widehat{D}_0=\{D_0(s):s\in {\mathbb R}\}\subseteq {\mathcal P}(\ell^2)$, 对每个$s\in {\mathbb R}$, $D_0(s)\subset\ell^2$是有界集.若对任意的$t\in {\mathbb R}$, $\widehat{D}=\{D(s):s\in {\mathbb R}\}\in {\mathcal D}_\sigma$, 存在一个$\tau_{0, \epsilon}(t, \widehat{D})\leqslant t$使得$U_\epsilon(t, \tau)D(\tau)\subseteq D_0(t)$对所有的$\tau\leqslant \tau_{0, \epsilon}(t, \widehat{D})$成立, 则称$\widehat{D}_0$是过程$\{U_\epsilon(t, \tau)\}_{t\geqslant \tau} $$\ell^2 上的有界拉回 {\mathcal D}_\sigma -吸收集. (2) 对任意给定的 t\in {{\Bbb R}} , \nu>0$$ \widehat{D}=\{D(s):s\in {\mathbb R}\}\in {\mathcal D}_\sigma$, 存在$M_{0, \epsilon}=M_{0, \epsilon}(t, \nu, \widehat{D})\in {\Bbb Z}_+ $$\tau_{0, \epsilon}=\tau_{0, \epsilon}(t, \nu, \widehat{D})\leqslant t 使得 \begin{eqnarray} \sup\limits_{u_{\tau^+}\in D(\tau)}\sum\limits_{|m|\geqslant M_{0, \epsilon}} |(U_\epsilon(t, \tau)u_{\tau^+})_m|^2 \leqslant \nu^2, \; \forall \, \tau\leqslant \tau_{0, \epsilon} \end{eqnarray} 成立, 则称过程 \{U_\epsilon(t, \tau)\}_{t\geqslant \tau} 具有拉回 {\mathcal D}_\sigma -渐近零性质. (3) 称集合族 \hat{{\cal A}}^\epsilon_{{\cal D}_\sigma} =\{{\cal A}^\epsilon_{{\cal D}_\sigma}(t):t\in{{\Bbb R}} \} \subseteq{\cal P}(\ell^2) 是过程 \{U_\epsilon(t, \tau)\}_{t\geqslant\tau} 的一个拉回 {\cal D}_\sigma -吸引子, 若满足 (a) 紧性:对任意的 t\in {{\Bbb R}} , {\cal A}^\epsilon_{{\cal D}_\sigma}(t)$$ \ell^2$上的一个非空紧集;

(b) 不变性: $U_\epsilon(t, \tau){\cal A}^\epsilon_{{\cal D}_\sigma}(\tau) ={\cal A}^\epsilon_{{\cal D}_\sigma}(t)$, $\forall \, \tau\leqslant t$;

(c) 拉回吸引性: $\hat{{\cal A}}^\epsilon_{{\cal D}_\sigma}$满足

记$R_\sigma(t)>0$使得

$\begin{eqnarray} R^2_\sigma(t) = 1+ \frac{{\rm e}^{-\sigma t}}{\kappa} \int_{-\infty}^t{\rm e}^{\sigma \theta}\|g(\theta)\|^2{\mathrm d}\theta, \; t\in {{\Bbb R}} , \end{eqnarray}$

$$$\frac{{\mathrm d}}{{\mathrm d} t} \sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u^\epsilon_m(t)|^2 +\kappa\sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u^\epsilon_m(t)|^2 \leqslant \frac{1}{\kappa}\sum\limits_{m\in {\Bbb Z}} \chi(\frac{|m|}{M})|g_m(t)|^2 +\frac{2\chi_0}{M}R^2_\sigma(t), \; t\neq\tau_j,$$$

$$$\frac{4\chi_0}{M}R^2_\sigma(t) \leqslant \sigma\nu^2/3, \; \forall\, M\geqslant M_1.$$$

$y(t)=\sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u^\epsilon_m(t)|^2$, $M\geqslant M_1$, 直接计算可得

$\begin{eqnarray} y(\tau_j^+) &=& \sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u^\epsilon_m(\tau_j^+)|^2 =\sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})\big|u^\epsilon_m(\tau_j)+\phi_{mj}(u^\epsilon_m(\tau_j))\big|^2 \\ &\leqslant& 2\sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u^\epsilon_m(\tau_j)|^2 +\sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|\phi_{mj}(u^\epsilon_m(\tau_j))|^2\\ &\leqslant& (2+2L^2)\sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u^\epsilon_m(\tau_j)|^2 =(2+2L^2)y(\tau_j), \; j\in {\mathbb Z}. \end{eqnarray}$

$\begin{eqnarray} y(\tau^+) = \sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u_m(\tau^+)|^2 \leqslant \|u(\tau^+)\|^2, \end{eqnarray}$

$\begin{eqnarray} \sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|u^\epsilon_m(t)|^2 &\leqslant& \|u_{\tau^+}\|^2(2+2L^2)^{n(\tau, t)}{\rm e}^{-\kappa(t-\tau)} {}\\ && + \int_\tau^t(2+2L^2)^{n[s, t)} {\rm e}^{-\kappa(t-s)} \bigg(\frac{1}{\kappa}\sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|g_m(s)|^2 +\frac{\sigma\nu^2}{3}\bigg){\mathrm d}s\\ &\leqslant& \|u_{\tau^+}\|^2{\rm e}^{-\sigma(t-\tau)} +\frac{{\rm e}^{-\sigma t}}{\kappa}\int_\tau^t{\rm e}^{\sigma s} \sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|g_m(s)|^2{\mathrm d}s {}\\ && +\frac{\nu^2}{3}, \; \, \forall\, t>\tau_1> \tau, M\geqslant M_1. \end{eqnarray}$

$$$\frac{{\rm e}^{-\sigma t}}{\kappa}\int_\tau^t{\rm e}^{\sigma s} \sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|g_m(s)|^2{\mathrm d}s \leqslant \frac{{\rm e}^{-\sigma t}}{\kappa}\int_{-\infty}^t{\rm e}^{\sigma s} \sum\limits_{m\in {\Bbb Z}}\chi(\frac{|m|}{M})|g_m(s)|^2{\mathrm d}s \leqslant \frac{\nu^2}{3}, \; \forall\, M\geqslant M_2.$$$

因为对任意的$\epsilon>0$, 过程$\{U_\epsilon(t, \tau)\}_{t\geqslant \tau} $$\ell^2 上连续.定理3.1的结论可以由引理3.1, 引理3.2和文献[43, 定理2.1]直接得到. ## 4 不变测度与统计解的构造 在这一节中, 我们证明对任意给定的 t \in {{\Bbb R}}$$ u_*\in\ell^2$, 映射$\tau\mapsto U_\epsilon(t, \tau)u_*$属于空间$PC((-\infty, t]; \ell^2)$; 且对任意的$\epsilon>0$, 这个$\ell^2$ -值映射是有界的.接着我们改进[32, 定理3.1]中的定义, 通过广义Banach极限和定理3.1中得到的拉回吸引子$\hat{{\mathcal A}^\epsilon}(t)$, 构造过程$\{U_\epsilon(t, \tau)\}_{t\geqslant \tau}$的一族Borel不变概率测度$\{{{\mathit{ m}}}^\epsilon_t\}_{t\in {{\Bbb R}} }$, 并且证明这族Borel概率不变测度$\{{{\mathit{ m}}}^\epsilon_t\}_{t\in {{\Bbb R}} }$只分段地满足Liouville型定理, 并且是脉冲离散Ginzburg-Landau方程组的统计解.

引理4.1的结论可由定理2.1和假设$\rm ({H2})$直接得到.事实上, 对任意给定的$t_*\in {{\Bbb R}} $$u_*\in\ell^2 , 有 \begin{eqnarray} \|u^\epsilon(t_*;\tau, u_*)\|^2 \leqslant \|u_*\|^2 +\frac{{\rm e}^{-\sigma t_*}}{\kappa} \int_{-\infty}^{t_*}{\rm e}^{\sigma \theta}\|g(\theta)\|^2{\mathrm d}\theta, \; \forall\, t_*>\tau. \end{eqnarray} 由假设 \rm ({H2}) 知上面不等式的右边是与 \tau 无关的有界量.证明完毕. 引理4.2 假设 \alpha , \epsilon , \kappa , \beta 都是正的常数, 1<\gamma<+\infty$$ \rm ({H1})–({H2})$成立.考虑给定的$\tau_*\in {{\Bbb R}} $$u_*\in\ell^2 , 则对于任意的 \epsilon>0 , 存在一个充分小的正数 \delta=\delta(\nu, \tau_*, u_*) , 使得 \begin{eqnarray} \|U_\epsilon(s, \tau)u_*-u_*\|<\nu, \; \forall\, \tau\in (\tau_*, \tau_*+\delta), \; \forall\, s\in (\tau, \tau_*+\delta). \end{eqnarray} 给定 \tau_*\in {{\Bbb R}}$$ \tau_*\in \ell^2$.不失一般性, 假设存在某个$j_0\in {\Bbb Z}$使得$\tau_*\in (\tau_{j_0}, \tau_{j_0+1}]$.我们分两种情形证明.

$\begin{eqnarray} \int_\tau^s\bigg\|\frac{{\mathrm d} U_\epsilon(\theta, \tau)u_*}{{\mathrm d}\theta}\bigg\|^2{\mathrm d}\theta \lesssim c_* = \|u_*\|^2 +\int_{\tau_*-d}^{\tau_*+d}\|g(\theta)\|^2{\mathrm d}\theta. \end{eqnarray}$

$\begin{eqnarray} \bigg \|\frac{{\mathrm d}U_\epsilon(\theta, \tau)u_*}{{\mathrm d}\theta}\bigg\|^2 \lesssim \|AU_\epsilon(\theta, \tau)u_*\|^2+\|U_\epsilon(\theta, \tau)u_*\|^2 +\|\tilde{f}(U_\epsilon(\theta, \tau)u_*)\|^2+\|g(\theta)\|^2. \end{eqnarray}$

$\begin{eqnarray} \|\tilde{f}(U_\epsilon(\theta, \tau)u_*)\|^2 \leqslant M^*_f\|U_\epsilon(\theta, \tau)u_*\|^2, \end{eqnarray}$

$\begin{eqnarray} \int_\tau^s\bigg\|\frac{{\mathrm d} U_\epsilon(\theta, \tau)u_*}{{\mathrm d}\theta}\bigg\|^2{\mathrm d}\theta &\lesssim& (1+M^*_f)\int_\tau^s\|U_\epsilon(\theta, \tau)u_*\|^2{\mathrm d}\theta +\int_\tau^s\|g(\theta)\|^2{\mathrm d}\theta\\ &\lesssim& \|u_*\|^2 +\int_\tau^s\|g(\theta)\|^2{\mathrm d}\theta \lesssim \|u_*\|^2 +\int_{\tau_*-d}^{\tau_*+d}\|g(\theta)\|^2{\mathrm d}\theta. \end{eqnarray}$

$\begin{eqnarray} \|U_\epsilon(s, \tau)u_*-u_*\|^2 &=& \big(U_\epsilon(s, \tau)u_*-u_*, U_\epsilon(s, \tau)u_*-u_*\big) \\ &=& \|U_\epsilon(s, \tau)u_*\|^2-\|u_*\|^2-2{\bf{Re}}\, \big(U_\epsilon(s, \tau)u_*-u_*, u_*\big) \\ &=& \int_\tau^s\frac{{\mathrm d} \|U_\epsilon(\theta, \tau)u_*\|^2}{{\mathrm d}\theta}{\mathrm d}\theta -2{\bf{Re}}\, \big(U_\epsilon(s, \tau)u_*-u_*, u_*\big). \end{eqnarray}$

$\begin{eqnarray} \int_{\tau}^{s}\frac{{\mathrm d} \|U_\epsilon(\theta, \tau)u_*\|^2}{{\mathrm d}\theta}{\mathrm d}\theta \lesssim \int_{\tau}^{s}\|g(\theta)\|^2{\mathrm d}\theta. \end{eqnarray}$

$\begin{eqnarray} \int_\tau^s\frac{{\mathrm d} \|U_\epsilon(\theta, \tau)u_*\|^2}{{\mathrm d}\theta}{\mathrm d}\theta \lesssim \int_{\tau}^{s}\|g(\theta)\|^2{\mathrm d}\theta < \frac{\nu^2}{2}, \; \tau_*<\tau\leqslant s\leqslant \tau_*+\delta'. \end{eqnarray}$

$\begin{eqnarray} \Big|\big(U_\epsilon(\theta, \tau)u_*-u_*, u_*\big)\Big| &=& \bigg|\bigg(\int_\tau^s \frac{{\mathrm d}U_\epsilon(\theta, \tau)u_*}{{\mathrm d}\theta}{\mathrm d}\theta, u_*\bigg)\bigg| \leqslant \|u_*\|\int_\tau^s\bigg\| \frac{{\mathrm d}U_\epsilon(\theta, \tau)u_*} {{\mathrm d}\theta}\bigg\|{\mathrm d}\theta \\ &\leqslant& \|u_*\|\bigg(\int_\tau^s\| \frac{{\mathrm d}U_\epsilon(\theta, \tau)u_*} {{\mathrm d}\theta}\|^2{\mathrm d}\theta \bigg)^{1/2}(s-\tau)^{1/2}{}\\ &\leqslant& c_*^{1/2}\|u_*\|(s-\tau)^{1/2}. \end{eqnarray}$

(4.10)式表明存在$\delta''=\delta''(\nu, \tau_*, u_*)\in (0, d)$使得

$\begin{eqnarray} \Big|\big(U_\epsilon(\theta, \tau)u_*-u_*, u_*\big)\Big| < \frac{\nu^2}{4}, \; \tau_*<\tau\leqslant s\leqslant \tau_*+\delta''. \end{eqnarray}$

$\delta=\min\{\delta', \delta''\}$, 则由(4.8), (4.10)和(4.11)式就得到(4.2)式. $\tau_*\in (\tau_{j_0}, \tau_{j_0+1})$的情形证明完毕.

(2) $U_\epsilon(t_*, \tau)u_* $$\tau\in (-\infty, t_*] , \tau\neq\tau_j\in (-\infty, t_*] , j\in {\Bbb Z} 处连续 \rm ; (3) U_\epsilon(t_*, \tau)u_* 在脉冲点 \tau_j\in (-\infty, t_*] , j\in {\Bbb Z} 处右极限存在. 首先证明(1).任意给定 s_*\in (-\infty, t_*] .我们证明 U_\epsilon(t_*, \tau)u_*$$ \tau=s_*$处左连续.事实上, 不失一般性, 假设对于某个$j_0\in {\Bbb Z} $$s_*\in (\tau_{j_0}, \tau_{j_0+1}] .对任意的 \tau\in (\tau_{j_0}, s_*] , 由过程的不变性质得 \begin{eqnarray} \|U_\epsilon(t_*, \tau)u_*-U_\epsilon(t_*, s_*)u_*\| = \|U_\epsilon(t_*, s_*)U(s_*, \tau)u_*-U_\epsilon(t_*, s_*)u_*\|. \end{eqnarray} 注意到 t_*$$ s_*$给定, $U_\epsilon(t_*, s_*): \ell^2\mapsto \ell^2$是连续映射.故$U_\epsilon(t_*, \tau)u_*$在点$\tau=s_*$处的连续性可由(4.12)和(4.13)式得到.

(2) 若通常的极限$\lim\limits_{t\rightarrow +\infty}\zeta(t)$存在, 则$\lim\limits_{t\rightarrow +\infty}\zeta(t) =\lim\limits_{t\rightarrow +\infty}\zeta(t)$.

$\begin{eqnarray} |\lim\limits_{t\rightarrow +\infty}\zeta(t)| \leqslant \limsup\limits_{t\rightarrow +\infty}|\zeta(t)|, \; \forall\, \zeta(\cdot)\in B_+, \end{eqnarray}$

$\begin{eqnarray} \lim\limits_{t\rightarrow -\infty}\zeta(t) =\lim\limits_{t\rightarrow +\infty}\zeta(-t). \end{eqnarray}$

$\begin{eqnarray} \frac{{\mathrm d} u^\epsilon}{{\mathrm d}t}= F^\epsilon(u^\epsilon, t)=-{\rm i}g(t)-(\epsilon-{\rm i}\alpha) Au^\epsilon-\kappa u^\epsilon+{\rm i}\beta\tilde{f}(u^\epsilon), \; \, t\neq \tau_j, \, j\in {\Bbb Z}. \end{eqnarray}$

$\begin{eqnarray} \frac{{\mathrm d}}{{\mathrm d}t}\Phi(u^\epsilon(t))=(\Phi'(u^\epsilon), F(u^\epsilon, t)), \; \, t\neq \tau_j, \, j\in {\Bbb Z}, \end{eqnarray}$

(a) 对任意的$u\in \ell^2$, $\rm Fréchet$导数$\Phi'(u)$存在, 即:对任意的$u\in \ell^2$, 存在$\Phi'(u)$使得

(b) 对任意的$u\in \ell^2$, 有$\Phi'(u)\in \ell^2$, 且作为$\ell^2 $$\ell^2 的映射, u\longmapsto \Phi'(u) 是连续且有界的 \rm ; (c) 方程(4.20)的每一个全局解 u^\epsilon(t) 都满足(4.21)式. 我们可以考虑文献[20, p178]中定义在 \ell^2 上的试验函数.考虑给定的 \varphi\in \ell^2 , \gamma$$ {\mathbb C}$上具有紧支集的连续可微的实值函数.对任意的$u\in \ell^2$, 定义

$\begin{eqnarray} \lim\limits_{t\rightarrow t_*^-}\int_{\ell^2} \psi(u^\epsilon){\mathrm d}{\mathit{ m}}^\epsilon_t(u^\epsilon) = \int_{\ell^2} \psi(u^\epsilon){\mathrm d}{\mathit{ m}}^\epsilon_{t_*}(u). \end{eqnarray}$

$t_*=\tau_{j+1}$, 我们应用与(4.26)式相同的证明得到$\int_{\ell^2} \psi(u^\epsilon){\mathrm d}{\mathit{ m}}^\epsilon_t(u^\epsilon)$在点$t_*$处的左连续性.为了证明$\lim\limits_{t\rightarrow t_*^+} \int_{\ell^2}\psi(u^\epsilon){\mathrm d}{\mathit{ m}}^\epsilon_t(u^\epsilon)$的存在性, 我们考虑$t_*<t'\leqslant t''<\tau_{j+2}$, 此时有

$$$\int_{\ell^2}\psi(u^\epsilon){\mathrm d}{\it {\mathit{ m}}}^\epsilon_{t''}(u^\epsilon) - \int_{\ell^2} \psi(u^\epsilon){\mathrm d}{\mathit{ m}}^\epsilon_{t'}(u^\epsilon) = \int_{{\mathcal A}^\epsilon(t')} \big(\psi(U_\epsilon(t'', t')u^\epsilon)-\psi(u^\epsilon)\big){\mathrm d}{\mathit{ m}}^\epsilon_{t'}(u_\epsilon).$$$

$\begin{eqnarray} \Psi(u)=\big(w, F(u, \cdot)\big), \quad u\in \ell^2, \end{eqnarray}$

$\Psi(\cdot): \ell^2\longmapsto {\mathbb R}$.下面证明$\Psi(\cdot)\in C(\ell^2)$.$u_*\in \ell^2$给定, 考虑$u\in \ell^2$, 其中$\|u_*-u\|\leqslant 1$.由(2.2), (2.11)和(4.20)式得到

$\begin{eqnarray} |\Psi(u_*)-\Psi(u)| &=& |\big(w, F^\epsilon(u_*, \cdot)-F^\epsilon(u, \cdot)\big)| \\ &\lesssim& |\big(w, A(u_*-u)\big)| +|(w, u_*-u)| +|\big(w, \tilde{f}(u_*)-\tilde{f}(u)\big)|\\ &\lesssim& M_f(u^*, u)\|w\|\|u_*-u\|, \end{eqnarray}$

(4.29)式表明由(4.28)式定义的实值函数$\Phi(\cdot) $$\ell^2 上连续.从(4.17)和(4.28)式可知映射 u\mapsto \big(w, F^\epsilon(u, \cdot)\big)=\Psi(u)$$ m^\epsilon_t$ -可积的.同时, 我们已经证明

${\mathbb R}$上是分段连续的且$\{\tau_j\}_{j\in {\Bbb Z}}$是其第一类间断点.因此它属于$L^1_{\rm loc}({\mathbb R})$.

$\begin{eqnarray} \|U_\epsilon(t, \tau) u_{\tau^+}\|^2 &\leqslant& \|u_{\tau^+}\|^2{\rm e}^{-\sigma(t-\tau)} +\frac{{\rm e}^{-\sigma t}}{\kappa} \int_\tau^t{\rm e}^{\sigma \theta}\|g(\theta)\|^2{\mathrm d}\theta\\ &\leqslant& G(t, \tau, \hat{D}){}\\ &:=& \sup\limits_{u_{\tau^+}\in D(\tau)}\|u_{\tau^+}\|^2 +\frac{{\rm e}^{-\sigma t}}{\kappa} \int_\tau^t{\rm e}^{\sigma \theta}\|g(\theta)\|^2{\mathrm d}\theta, \; \forall\, t>\tau, \, \forall\, \epsilon\in [0, 1), \end{eqnarray}$

$\begin{eqnarray} \left\{ \begin{array}{ll} {\rm i}\frac{{\mathrm d} v}{{\mathrm d}t}+\alpha Av+{\rm i}\epsilon Au^\epsilon+{\rm i}\kappa v +\beta|u^\epsilon|^{2\gamma}u^\epsilon -\beta|u|^{2\gamma}u=0, \; t>\tau, \, t\neq \tau_j, \, j\in {\Bbb Z}, \\ v(\tau_j^+)-v(\tau_j)=\phi_j(u^\epsilon(\tau_j))-\phi_j(u(\tau_j)), \; \, j\in {\Bbb Z}, \\ v(\tau^+)=0, \; \tau\in {{\Bbb R}} . \end{array} \right. \end{eqnarray}$

$-{\rm i}v$与方程(5.3)作内积并取实部, 可得

$$$\frac 12\frac{{\mathrm d}}{{\mathrm d}t}\|v\|^2 +{\bf{Re}}\, \epsilon(B u^\epsilon, B v) +\kappa\|v\|^2 +{\bf{Im}}\, \beta(|u^\epsilon|^{2\gamma}u^\epsilon -|u|^{2\gamma}u, \bar{v}) =0, \; t>\tau, \, t\neq \tau_j, \, j\in {\Bbb Z}.$$$

$$$\big|{\bf{Re}}\, \epsilon(B u^\epsilon, B v)\big| \leqslant 4\epsilon\|u^\epsilon\|\|v\| \leqslant \frac{\kappa}{4}\|v\|^2 +\frac{32\epsilon^2}{\kappa}\|u^\epsilon\|^2.$$$

$\begin{eqnarray} \big|{\bf{Im}}\, \beta(|u^\epsilon|^{2\gamma}u^\epsilon -|u|^{2\gamma}u, \bar{v})\big| \leqslant \frac{\kappa}{4}\|v\|^2 +\frac{2\beta^2}{\kappa}\sum\limits_{m\in {\Bbb Z}} \big||u^\epsilon_m|^{2\gamma}u^\epsilon_m -|u_m|^{2\gamma}u_m\big|^2 \end{eqnarray}$

$\begin{eqnarray} && \sum\limits_{m\in {\Bbb Z}}\big||u^\epsilon_m|^{2\gamma}u^\epsilon_m -|u_m|^{2\gamma}u_m\big|^2{}\\ &=& \sum\limits_{m\in {\Bbb Z}}\big|f(|u^\epsilon_m|^2)u^\epsilon_m-f(|u_m|^2)u_m\big|^2 \\ &\leqslant& 2\sum\limits_{m\in {\Bbb Z}}\big|f(|u^\epsilon_m|^2)\big|^2|u^\epsilon_m-u_m\big|^2 + 2\sum\limits_{m\in {\Bbb Z}}\big|f(|u^\epsilon_m|^2)-f(|u_m|^2)|^2|u_m|^2 \\ &\leqslant& 2f^2(\|u^\epsilon\|^2)\|v\|^2 + 2\|u\|^2\underline{\sum\limits_{m\in {\Bbb Z}}\big|f(|u^\epsilon_m|^2)-f(|u_m|^2)|^2}. \end{eqnarray}$

$\begin{eqnarray} && \sum\limits_{m\in {\Bbb Z}}\big|f(|u^\epsilon_m|^2)-f(|u_m|^2)|^2{}\\ &=& \sum\limits_{m\in {\Bbb Z}}\big|f'(\theta|u^\epsilon_m|^2+(1-\theta)|u_m|^2)|^2 \big||u^\epsilon_m|+|u_m|\big|^2\big||u^\epsilon_m|-|u_m|\big|^2\\ &\leqslant& \gamma^2\sum\limits_{m\in {\Bbb Z}}\big|f'(\|u^\epsilon\|^2+\|u\|^2)|^2 \big(|u^\epsilon_m|^2+|u_m|^2\big)\big||u^\epsilon_m|-|u_m|\big|^2\\ &\leqslant& \gamma^2\big|f'(\|u^\epsilon\|^2+\|u\|^2)|^2\big(\|u^\epsilon\|^2+\|u\|^2\big)\|v\|^2. \end{eqnarray}$

$\begin{eqnarray} \|u^\epsilon\|^2+\|u\|^2 \leqslant \bar{G}=\bar{G}(T, \tau, \widehat{D}) := 2\sup\limits_{\tau\leqslant t\leqslant T}G(t, \tau, \widehat{D}). \end{eqnarray}$

$\begin{eqnarray} \frac{{\mathrm d}}{{\mathrm d}t}\|v\|^2 +\bigg(\kappa -\frac{8\beta^2\bar{G}^{4\gamma}}{\kappa} -4\gamma^2\bar{G}^{2\gamma-1}\bigg)\|v\|^2 \lesssim \bar{G}\epsilon^2, \; \tau<t\leqslant T, \, t\neq \tau_j, \, j\in {\Bbb Z}. \end{eqnarray}$

$\begin{eqnarray} \|v(\tau_j^+)\|^2 &=& \sum\limits_{m\in {\Bbb Z}}|v_m(\tau_j^+)|^2{}\\ &=& \sum\limits_{m\in {\Bbb Z}}\big |v_m(\tau_j) +\phi_{mj}(u^{\epsilon}_m(\tau_j))-\phi_{mj}(u_m(\tau_j))\big|^2 \\ &\leqslant& 2\sum\limits_{m\in {\Bbb Z}}|v_m(\tau_j)|^2 + 2\sum\limits_{m\in {\Bbb Z}}\big|\phi_{mj}(u^{\epsilon}_m(\tau_j)) -\phi_{mj}(u_m(\tau_j))\big|^2 \\ &\leqslant& 2\sum\limits_{m\in {\Bbb Z}}|v_m(\tau_j)|^2 + 2\sum\limits_{m\in {\Bbb Z}}L^2 \big|u^{\epsilon}_m(\tau_j)-u_m(\tau_j)\big|^2{}\\ &=& (2+2L^2)\|v(\tau_j)\|^2. \end{eqnarray}$

$\begin{eqnarray} \frac{{\mathrm d} u}{{\mathrm d}t}= F(u, t):=-{\rm i}g(t)+{\rm i}\alpha Au-\kappa u+{\rm i}\beta\tilde{f}(u), \; \, t\neq \tau_j, \, j\in {\Bbb Z}. \end{eqnarray}$

$\begin{eqnarray} &&\lim\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int_\tau^t \psi\big(U(t, s)v(s)\big){\mathrm d}s {}\\ &=& \int_{{\mathcal A}(t)}\psi(u){\mathrm d}{\textsf m}_t(u) = \int_{\ell^2}\psi(u){\mathrm d}{\textsf m}_t(u) \end{eqnarray}$

$\begin{eqnarray} &=& \lim\limits_{\tau\rightarrow -\infty}\frac{1}{t-\tau}\int_\tau^t\int_{\ell^2} \psi\big(U(t, s)u\big){\mathrm d}{\textsf m}_s(u){\mathrm d}s. \end{eqnarray}$

$\begin{eqnarray} \lim\limits_{\epsilon\rightarrow 0^+} \int_{\ell^2}\psi(u){\mathrm d}{\textsf m}^\epsilon_t(u) = \int_{\ell^2}\psi(u){\mathrm d}{\textsf m}_t(u), \; \forall\, \psi\in C(\ell^2), \forall\, \, t\in {{\Bbb R}} . \end{eqnarray}$

令$v(\cdot):{\mathbb R}\mapsto \ell^2$是一个连续映射且满足$v(\cdot)\in {\mathcal D}_\sigma$.对于每一个$t\in {{\Bbb R}}$和任意的$\tau<t$, 由引理5.2可得

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