¹伊犁师范大学应用数学研究所，新疆 伊犁

²伊犁师范大学数学与统计学院，新疆 伊犁

收稿日期：2022年6月18日；录用日期：2022年7月20日；发布日期：2022年7月27日

摘要

本文研究了带有乘性噪声的非自治随机时滞Kuramoto-Sivashinsky方程解的长时间行为。通过对解的一致估计，结合随机吸引子的存在性定理，证明了由该方程 $C\left(\left[-\rho ,0\right],L^2\left(I\right)\right)$ 所生成的随机动力系统吸引子的存在性。

关键词

时滞随机Kuramoto-Sivashinsky方程，随机吸引子，渐近紧性，乘性噪声

Pullback Attractors for Delay Non-Autonomous Stochastic Kuramoto-Sivashinsky Equation with Multiplicative Noise

Zhiheng Xu¹, Yucun Diao²

¹Institute of Applied Mathematics, Yili Normal University, Yili Xinjiang

²School of Mathematics and Statistics, Yili Normal University, Yili Xinjiang

Received: Jun. 18^th, 2022; accepted: Jul. 20^th, 2022; published: Jul. 27^th, 2022

ABSTRACT

In this paper, we study the long time behavior of delay non-autonomous stochastic Kuramoto-Sivashinsky equation with multiplicative noise. We prove the existence of pullback attractors in the space $C\left(\left[-\rho ,0\right],L^2\left(I\right)\right)$ for the dynamical system generated by the equation above, by some uniform estimation, together with the existence theorem of pullback attractors.

Keywords:Delay Non-Autonomous Stochastic Kuramoto-Sivashinsky Equation, Pullback Attractors, Asymptotic Compactness, Multiplicative Noise

Copyright © 2022 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

D-拉回随机吸引子最早是由 [1] [2] 提出。其中 [1] 探讨了时滞随机抛物方程的拉回吸引子的存在性及其上半连续性，并为拉回吸引子的存在性提供了充分条件。 [2] 提供了充要条件。本文则是在文献 [3] [4] [5] 的基础上，讨论了带乘法噪声的时滞非自治随机Kuramoto-Sivanshinsky方程的拉回吸引子。

考虑如下带有时滞的非自治随机Kuramoto-Sivashinsky方程(1)的初边值问题：

$\begin{array}{l}\text{d}u+\left(\alpha D^{4}u+D^{2}u+uDu\right)\text{d}t=\left(f\left(t,x\right)+g\left(u\left(t-\rho \right),x\right)\right)\text{d}t+u\text{d}W,t>\tau ,x\in I\\ u\left(\tau +\sigma ,x\right)=u_{\tau }\left(\sigma ,x\right)=\phi \left(\sigma ,x\right)\\ D^{i}u\left(t,\frac{-l}{2}\right)=D^{i}u\left(t,\frac{l}{2}\right),i=0,1,2,3\\ {\displaystyle {\int }_{I}u\left(t,x\right)\text{d}x}=0,\forall t\in R\end{array}$

其中f是非线性项，g是时滞项，W是定义在概率空间 $\left(\Omega ,F,P\right)$ 上的双边实值Wiener过程。ρ > 0是系统的时滞，常数α满足

$\alpha >\frac{l^2}{4{\pi }^2}$

2. 连续随机动力系统

本节讨论了由方程(1)时滞随机Kuramoto-Sivanshinsky方程所生成的连续随机动力系统。

首先做一些书写符号上的约定。用 $\Vert \cdot \Vert$ 和 $\left(\cdot ,\cdot \right)$ 分别表示 $L^2\left(I\right)$ 空间上的范数和内积， ${\Vert \cdot \Vert }_{H_{\rho }}$ 表示空间 $H_{\rho }$ 上的范数。

对外力项f与时滞项g做如下假设：

H₁. ${\displaystyle {\int }_{-\infty }^{\tau }{\text{e}}^{\beta \left(r-\tau \right)}\left|f\left(r\right)\right|\text{d}r},\forall \beta >0,\tau \in R$

H₂. $g\left(0,x\right)=0,\forall x\in I;\left|g\left(s_1,x\right)-g\left(s_2,x\right)\right|\le L_g\left|s_1-s_2\right|,\forall x\in I$

令 $v\left(t,\tau ,\omega ,\varphi \right)={\text{e}}^{-z\left({\theta }_t\omega \right)}u\left(t,\tau ,\omega ,\phi \right)$

则 $u\left(t,\tau ,\omega ,\varphi \right)={\text{e}}^{z\left({\theta }_t\omega \right)}v\left(t,\tau ,\omega ,\phi \right)$

其中 $z\left({\theta }_t\omega \right)=-{\displaystyle {\int }_{-\infty }^0{\text{e}}^r{\theta }_t\omega \left(r\right)\text{d}r}$

由于 $\frac{1}{2}{u}^2\text{d}t+u\text{d}W=u\circ W$，这里“ $\circ$ ”表示Strotnnovitch积分，则代入可得(2)

$\begin{array}{l}\frac{\partial v}{\partial t}+\alpha D^{4}v+D^{2}v+{\text{e}}^{z\left({\theta }_t\omega \right)}vDv={\text{e}}^{-z\left({\theta }_t\omega \right)}f\left(t,x\right)+{\text{e}}^{-z\left({\theta }_t\omega \right)}g\left({\text{e}}^{z\left({\theta }_{t-\rho }\omega \right)}v\left(t-\rho \right),x\right)+z\left({\theta }_t\omega \right)v\\ v\left(\tau +\sigma ,x\right)=\varphi \left(\sigma ,x\right),\sigma \in \left[-\rho ,0\right]\\ D^{i}v\left(t,\frac{-l}{2}\right)=D^{i}v\left(t,\frac{l}{2}\right),i=0,1,2,3\\ {\displaystyle {\int }_{I}v\left(t,x\right)\text{d}x}=0,\forall t\in R\end{array}$

由Galerkin逼近法(见参考文献 [6])，通过假设H₁，H₂可得对任意 $\tau \in R,\omega \in \Omega ,\varphi \in C\left(\left[-\rho ,0\right],L^2\left(I\right)\right)$，方程(2)存在唯一解 $v\left(·,\tau ,\omega ,\varphi \right)$。因此可以定义一连续随机动力系统 [1]。

$\Phi \left(t,\tau ,\omega ,\varphi \right)(\cdot )=u_{t+\tau }\left(\cdot ,\tau ,{\theta }_{-\tau }\omega ,\varphi \right)$

且存在正常数 $\eta$ 使得

$\alpha {\Vert D^{2}v\Vert }^2-{\Vert Dv\Vert }^2\ge \eta \left({\Vert D^{2}v\Vert }^2+{\Vert v\Vert }^2\right),\forall v\in H_{per}^2(\; I\; )$

定义随机变量

$\begin{array}{l}\gamma \left(\omega \right)=\eta -2\left|z\left(\omega \right)\right|-{\text{e}}^{\frac{1}{2}\eta \rho }{L}_g\left({\text{e}}^{2z\left(\omega \right)}+{\text{e}}^{-2z\left(\omega \right)}\right)\\ {\gamma }^{\text{*}}\left(\omega \right)=\eta -2E\left(\left|z\right|\right)-{\text{e}}^{\frac{1}{2}\eta \rho }{L}_g\left[E\left({\text{e}}^{2z}\right)+E\left({\text{e}}^{-2z}\right)\right]\end{array}$

则由遍历定理可得

$\underset{t\to \infty }{\lim }\frac{1}{t}{\displaystyle {\int }_0^t\gamma \left({\theta }_r\omega \right)}=E\left(r\right)={\gamma }^{\text{*}}$

存在Ω的不变子集使得

$\begin{array}{l}\underset{t\to \infty }{\lim }\frac{z\left({\theta }_t\omega \right)}{t}=\underset{t\to \infty }{\lim }\frac{1}{t}{\displaystyle {\int }_0^{t}z\left({\theta }_r\omega \right)}=0\\ \underset{t\to \infty }{\lim }\frac{1}{t}{\displaystyle {\int }_0^t\left|z\left({\theta }_r\omega \right)\right|\text{d}r}=E\left(\left|z\right|\right)=\frac{1}{\sqrt{\pi }}\end{array}$

若集合D满足

$\underset{t\to +\infty }{\lim }{\text{e}}^{-rt}\underset{s\le \tau }{\sup }{\Vert \mathcal{D}\left(s-t,{\theta }_{-\tau }\omega \right)\Vert }^2=0,\forall r,\tau \in R,\omega \in \Omega$

则称D为后项缓增集。

3. 一致估计

为得到随机动力系统拉回吸收集的存在性，首先在空间 $C\left(\left[-\rho ,0\right],L^2\left(I\right)\right)$ 上进行估计。有如下引理

引理1 若假设H₁，H₂成立，则存在 $T=T\left(\tau ,\omega ,D\right)>0$，使得当t > T时，有

$\begin{array}{l}{\Vert v\left(\tau +\sigma ,\tau -t,{\theta }_{-\tau }\omega ,\varphi \right)\Vert }^2\le R\left(\tau ,\omega \right)+c{\displaystyle {\int }_{-\infty }^0{\text{e}}^{{\displaystyle {\int }_0^{r}r\left({\theta }_l\omega \right)\text{d}l}}{\text{e}}^{-2z\left({\theta }_r\omega \right)}{\Vert f\left(r+\tau \right)\Vert }^2\text{d}r}\\ {\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert D^{2}v\left(r\right)\Vert }^2\text{d}r}\le C\left(\omega \right)(R\left(\tau ,\omega \right)+M(\; \tau \; ))\end{array}$

其中 $R\left(\tau ,\omega \right)$ 与 $M\left(\tau \right)$ 由如下(3)式定义

$\begin{array}{l}R\left(\tau ,\omega \right)={\displaystyle {\int }_{-\infty }^0{\text{e}}^{{\displaystyle {\int }_0^{r}r\left({\theta }_l\omega \right)\text{d}l}}{\text{e}}^{-2z\left({\theta }_r\omega \right)}{\Vert f\left(r+\tau \right)\Vert }^2\text{d}r}\\ M\left(\tau \right)={\displaystyle {\int }_{-\infty }^{\tau }{\text{e}}^{\eta \left(r-\tau \right)}{\Vert f\left(r\right)\Vert }^2\text{d}r}\end{array}$

证明：将(2)与 $v\left(r,\tau -t,\theta -\tau \omega ,\varphi \right)$ 在空间 $L^2\left(I\right)$ 上做内积，并注意到 $\left(vDv,v\right)=0$，得

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}r}{\Vert v\left(r\right)\Vert }^2+\alpha {\Vert D^{2}v\left(r\right)\Vert }^2-{\Vert Dv\left(r\right)\Vert }^2\\ ={\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(f\left(r\right),v\right)+{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(g\left({\text{e}}^{z\left({\theta }_{r-\rho -\tau }\omega \right)}v\left(r-\rho \right)\right),v\right)+z\left({\theta }_{r-\tau }\omega \right){\Vert v\left(r\right)\Vert }^2\end{array}$

由Young不等式

${\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(f\left(r\right),v\right)\le \frac{\eta }{2}{\Vert v\left(r\right)\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2$

代入并整理，得

$\begin{array}{l}\frac{\text{d}}{\text{d}r}{\Vert v\left(r\right)\Vert }^2+2\eta {\Vert D^{2}v\left(r\right)\Vert }^2+\eta {\Vert v\left(r\right)\Vert }^2\le 2{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\\ \left(g\left({\text{e}}^{z\left({\theta }_{r-\rho -\tau }\omega \right)}v\left(r-\rho \right)\right),v\right)+2z\left({\theta }_{r-\tau }\omega \right){\Vert v\left(r\right)\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2\end{array}$

不等式左右两端同乘 ${\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}>0$，可得

$\begin{array}{c}\frac{\text{d}}{\text{d}r}{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\Vert v\left(r\right)\Vert }^2\le 2{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(g\left({\text{e}}^{z\left({\theta }_{r-\rho -\tau }\omega \right)}v\left(r-\rho \right)\right),v\right)\\ +\left(2z\left({\theta }_{r-\tau }\omega \right){\Vert v\left(r\right)\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2\right){\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}\end{array}$

在区间 $\left[\tau -t,\tau +\sigma \right],\sigma \in \left[-2\rho -2,0\right]$ 上积分，可得

$\begin{array}{l}{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\Vert v\left(\tau +\sigma \right)\Vert }^2\\ \le 2{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(g\left({\text{e}}^{z\left({\theta }_{r-\rho -\tau }\omega \right)}v\left(r-\rho \right)\right),v\right)\text{d}r}\\ +{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }\left(2z\left({\theta }_{r-\tau }\omega \right){\Vert v\left(r\right)\Vert }^2\text{d}r+c{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2}\right)}+{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}\text{d}r+{\Vert \varphi \Vert }_{H_{\rho }}^2\end{array}$

其中

$\begin{array}{l}2{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(g\left(v\left(r-\rho \right){\text{e}}^{z\left({\theta }_{r-\tau -\rho }\omega \right)}\right),v\right)\text{d}r}\\ \le {\text{e}}^{\frac{\eta }{2}\rho }{L}_g{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}{\Vert v\left(r\right)\Vert }^2\text{d}r}\\ +{\text{e}}^{-\frac{\eta }{2}\rho }\frac{1}{L_g}{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\Vert g\left(v\left(r-\rho \right)\right){\text{e}}^{z\left({\theta }_{r-\tau -\rho }\omega \right)}\Vert }^2\text{d}r}\\ \le {\text{e}}^{\frac{\eta }{2}\rho }{L}_g{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}{\Vert v\left(r\right)\Vert }^2\text{d}r}\\ +L_g{\text{e}}^{-\frac{\eta }{2}\rho }{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r-\rho }r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{{\displaystyle {\int }_{r-\rho }^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{2z\left({\theta }_{r-\rho -\tau }\omega \right)}{\Vert v\left(r-\rho \right)\Vert }^2\text{d}r}\end{array}$

$\begin{array}{l}\le {\text{e}}^{\frac{\eta }{2}\rho }{L}_g{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}{\Vert v\left(r\right)\Vert }^2\text{d}r}\\ +L_g{\text{e}}^{\frac{n}{2}\rho }{\displaystyle {\int }_{\tau -t-\rho }^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{2z\left({\theta }_{r-\rho }\omega \right)}{\Vert v\left(r-\rho \right)\Vert }^2\text{d}r}\\ \le {\text{e}}^{\frac{\eta }{2}\rho }{L}_g{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}{\Vert v\left(r\right)\Vert }^2\text{d}r}\\ +L_g{\text{e}}^{\frac{n}{2}\rho }{\displaystyle {\int }_{\tau -t-\rho }^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau -t}^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{2z\left({\theta }_{r-\rho }\omega \right)}{\Vert \varphi \Vert }_{H_{\rho }}^2\text{d}r}\end{array}$

由 $\gamma \left(\omega \right)$ 的定义，可知

$2z\left({\theta }_{r-\tau }\omega \right)-\eta +r\left({\theta }_{l-\tau }\omega \right)+L_g{\text{e}}^{\frac{n}{2}\rho }\left({\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}+{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}\right)\le 0$

上述不等式左右两端同时除以 ${\text{e}}^{{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}>0$，可得

$\begin{array}{l}{\Vert v\left(\tau +\sigma ,\tau -t,{\theta }_{-\tau }\omega ,\varphi \right)\Vert }^2\\ \le c\left({\displaystyle {\int }_{\tau -t-\rho }^{\tau -t}{\text{e}}^{{\displaystyle {\int }_{\tau +\sigma }^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}\text{d}r}+{\text{e}}^{-{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}\right){\Vert \varphi \Vert }_{H_{\rho }}^2\\ +c{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\tau +\sigma }^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2\text{d}r}\end{array}$

$\begin{array}{l}=c\left({\displaystyle {\int }_{\tau -t-\rho }^{\tau -t}{\text{e}}^{{\displaystyle {\int }_{\sigma }^{r-\tau }r\left({\theta }_l\omega \right)\text{d}l}}{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}\text{d}r}+{\text{e}}^{-{\displaystyle {\int }_{-t}^{\sigma }r\left({\theta }_l\omega \right)\text{d}l}}\right){\Vert \varphi \Vert }_{H_{\rho }}^2\\ +c{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }{\text{e}}^{{\displaystyle {\int }_{\sigma }^{r-\tau }r\left({\theta }_l\omega \right)\text{d}l}}{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2\text{d}r}\\ \le c\left({\displaystyle {\int }_{-t-\rho }^{-t}{\text{e}}^{{\displaystyle {\int }_0^{r}r\left({\theta }_l\omega \right)\text{d}l}}{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}\text{d}r}+{\text{e}}^{-{\displaystyle {\int }_{-t}^{0}r\left({\theta }_l\omega \right)\text{d}l}}\right){\Vert \varphi \Vert }_{H_{\rho }}^2\\ +c{\displaystyle {\int }_{-\infty }^0{\text{e}}^{{\displaystyle {\int }_0^{r}r\left({\theta }_l\omega \right)\text{d}l}}{\text{e}}^{-2z\left({\theta }_r\omega \right)}{\Vert f\left(r+\tau \right)\Vert }^2\text{d}r}\end{array}$

当 $t\to \infty$ 时

$c\left({\displaystyle {\int }_{\tau -t-\rho }^{\tau -t}{\text{e}}^{{\displaystyle {\int }_{\tau +\sigma }^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}\text{d}r}+{\text{e}}^{-{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}\right){\Vert \varphi \Vert }_{H_{\rho }}^2\to 0$

故存在T > 0，使得当t > T时，成立

$c\left({\displaystyle {\int }_{\tau -t-\rho }^{\tau -t}{\text{e}}^{{\displaystyle {\int }_{\tau +\sigma }^{r}r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}\text{d}r}+{\text{e}}^{-{\displaystyle {\int }_{\tau -t}^{\tau +\sigma }r\left({\theta }_{l-\tau }\omega \right)\text{d}l}}\right){\Vert \varphi \Vert }_{H_{\rho }}^2\le R\left(\tau ,\omega \right)$

从而

$\underset{\sigma \in \left[-2\rho -2,0\right]}{\sup }{\Vert v\left(\tau +\sigma ,\tau -t,{\theta }_{-\tau }\omega ,\varphi \right)\Vert }^2\le R\left(\tau ,\omega \right)$

又由

$\begin{array}{c}{\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert D^{2}v\left(r\right)\Vert }^2\text{d}r}\le c\left(\omega \right){\displaystyle {\int }_{\tau -\rho -2}^{\tau }\left(g\left(v\left(r-\rho \right){\text{e}}^{z\left({\theta }_{r-\tau -\rho }\omega \right)}\right),v\right)\text{d}r}+c\left(\omega \right){\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert f\left(r\right)\Vert }^2\text{d}r}\\ +c\left(\omega \right){\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert v\left(r\right)\Vert }^2\text{d}r}+{\Vert v\left(\tau -\rho -2\right)\Vert }^2\end{array}$

对不等式右端第一项进行放缩

$\begin{array}{l}c\left(\omega \right){\displaystyle {\int }_{\tau -\rho -2}^{\tau }\left(g\left(v\left(r-\rho \right){\text{e}}^{z\left({\theta }_{r-\tau -\rho }\omega \right)}\right),v\right)\text{d}r}\\ \le \frac{1}{2}{\displaystyle {\int }_{\tau -\rho -2}^{\tau }{L}_g^2{\text{e}}^{2z\left({\theta }_{r-\tau -\rho }\omega \right)}{\Vert v\left(r-\rho \right)\Vert }^2\text{d}r}+\frac{1}{2}{\displaystyle {\int }_{\tau -\rho -r}^{\tau }{\Vert v\left(r\right)\Vert }^2\text{d}r}\\ \le c\left(\omega \right)\underset{\sigma \in \left[-2\rho -2,0\right]}{\sup }{\Vert v\left(\tau +\sigma \right)\Vert }^2\left(\rho +2\right)+\frac{1}{2}\left(\rho +2\right)\underset{\sigma \in \left[-\rho -2,0\right]}{\sup }{\Vert v\left(r\right)\Vert }^2\\ \le cR\left(\tau ,\omega \right)\end{array}$

对不等式右端第二项进行放缩

$\begin{array}{c}c\left(\omega \right){\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert f\left(r\right)\Vert }^2\text{d}r}\le {\text{e}}^{\eta \left(\rho +2\right)}{\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\text{e}}^{\eta \left(r-\tau \right)}{\Vert f\left(r\right)\Vert }^2\text{d}r}\\ \le {\text{e}}^{\eta \left(\rho +2\right)}{\displaystyle {\int }_{-\infty }^{\tau }{\Vert f\left(r\right)\Vert }^2\text{d}r}\\ ={\text{e}}^{\eta \left(\rho +2\right)}M(\; \tau \; )\end{array}$

同理对不等式右端第二项进行放缩

$\begin{array}{l}{\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert v\left(r\right)\Vert }^2\text{d}r}+{\Vert v\left(\tau -\rho -2\right)\Vert }^2\\ \le \left(\rho +2\right)\underset{\sigma \in \left[-\rho -2,0\right]}{\sup }{\Vert v\left(\tau +\sigma \right)\Vert }^2+{\Vert v\left(\tau -\rho -2\right)\Vert }^2\le cR\left(\tau ,\omega \right)\end{array}$

于是

${\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert D^{2}v\left(r\right)\Vert }^2\text{d}r}\le c\left(\omega \right)(R\left(\tau ,\omega \right)+M(\; \tau \; ))$

此即为所证。

下将在空间 $H_0^1\left(I\right)$ 进行估计，再由 $H_0^1\left(I\right)$ 紧嵌入 $L^2\left(I\right)$，得到随机动力系统在空间 $L^2\left(I\right)$ 上的渐近紧性(见参考文献 [7])。

引理2 若假设H₁，H₂成立，则存在T > 0，使得当t > T时，有

${\displaystyle {\int }_{\tau -\rho -1}^{\tau }{\Vert D^{2}v\left(r\right)\Vert }^2\text{d}r}\le C\left(\omega \right)\left(1+R\left(\tau ,\omega \right)\right)(R\left(\tau ,\omega \right)+M(\; \tau \; ))$

其中 $R\left(\tau ,\omega \right)$ 与 $M\left(\tau \right)$ 由(3)定义。

证明：

$\begin{array}{l}\frac{1}{2}\frac{\text{d}}{\text{d}r}{\Vert D^{2}v\left(r\right)\Vert }^2+\alpha {\Vert D^{4}v\left(r\right)\Vert }^2+\left(D^{2}v,D^4\right)+{\text{e}}^{z\left({\theta }_{r-\tau }\omega \right)}\left(vDv,D^{4}v\right)\\ ={\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(f\left(r\right),D^{4}v\right)+{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(g\left(v\left(r-\rho \right){\text{e}}^{z\left({\theta }_{r-\rho -\tau }\omega \right)}\right),D^{4}v\right)+z\left({\theta }_{r-\tau }\omega \right){\Vert D^{2}v\left(r\right)\Vert }^2\end{array}$

由Young不等式，可得

$-\left(D^{2}v,D^4\right)+{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(f\left(r\right),D^{4}v\right)\le \frac{\alpha }{6}{\Vert D^{4}v\left(r\right)\Vert }^2+c{\Vert D^{2}v\left(r\right)\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2$

由Agmon不等式，可得

${\text{e}}^{z\left({\theta }_{r-\tau }\omega \right)}\left(vDv,D^{4}v\right)\le \frac{\alpha }{6}{\Vert D^{4}v\left(r\right)\Vert }^2+c{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}{\Vert v\Vert }^2{\Vert D^{2}v\left(r\right)\Vert }^2$

由假设H₂可得

${\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(g\left(v\left(r-\rho \right){\text{e}}^{z\left({\theta }_{r-\rho -\tau }\omega \right)}\right),D^{4}v\right)\le \frac{\alpha }{6}{\Vert D^{4}v\left(r\right)\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\text{e}}^{2z\left({\theta }_{r-\tau -\rho }\omega \right)}{\Vert v\left(r-\rho \right)\Vert }^2$

所以

$\begin{array}{c}\frac{\text{d}}{\text{d}r}{\Vert D^{2}v\left(r\right)\Vert }^2+\gamma {\Vert D^{2}v\left(r\right)\Vert }^2\le c{\Vert D^{2}v\left(r\right)\Vert }^2+c{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}{\Vert v\Vert }^2{\Vert D^{2}v\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2\\ +c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}c{\text{e}}^{2z\left({\theta }_{r-\tau -\rho }\omega \right)}{\Vert v\left(r-\rho \right)\Vert }^2+cz\left({\theta }_{r-\tau }\omega \right){\Vert D^{2}v\left(r\right)\Vert }^2\end{array}$

将上式在 $r\in \left[s,\tau +\sigma \right],\sigma \in \left[-\rho -1,0\right],s\in \left[\tau +\sigma -1,\tau +\sigma \right]$ 上积分，

再在 $s\in \left[\tau +\sigma -1,\tau +\sigma \right]$ 上积分，可得

${\Vert D^{2}v\left(\tau +\sigma \right)\Vert }^2\le c\left(\omega \right){\displaystyle {\int }_{\tau -\rho -2}^{\tau }\left({\Vert D^{2}v\left(r\right)\Vert }^2+{\Vert v\left(r\right)\Vert }^2{\Vert D^{2}v\left(r\right)\Vert }^2+{\Vert f\left(r\right)\Vert }^2+{\Vert v\left(r-\rho \right)\Vert }^2\right)\text{d}r}$

注意到上述不等式右端第二项

${\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert v\left(r\right)\Vert }^2{\Vert D^{2}v\left(r\right)\Vert }^2\text{d}r}\le \underset{\sigma \in \left[-\rho -2,0\right]}{\sup }{\Vert v\left(\sigma \right)\Vert }^2{\displaystyle {\int }_{\tau -\rho -2}^{\tau }{\Vert D^{2}v\left(r\right)\Vert }^2\text{d}r}$

于是

$\underset{\sigma \in \left[-\rho -1,0\right]}{\sup }{\Vert D^2\left(\tau +\sigma \right)\Vert }^2\le C\left(\omega \right)\left(1+R\left(\tau ,\omega \right)\right)(R\left(\tau ,\omega \right)+M(\; \tau \; ))$

在 $\left[\tau -\rho -1,\tau \right]$ 上积分，可得

$\begin{array}{l}{\displaystyle {\int }_{\tau -\rho -1}^{\tau }{\Vert D^{4}v\left(r\right)\Vert }^2\text{d}r}\\ \le {\Vert D^2\left(\tau -\rho -1\right)\Vert }^2+c\left(\omega \right){\displaystyle {\int }_{\tau -\rho -1}^{\tau }\left({\Vert D^{2}v\left(r\right)\Vert }^2+{\Vert v\left(r\right)\Vert }^2{\Vert D^{2}v\left(r\right)\Vert }^2+{\Vert f\left(r\right)\Vert }^2+{\Vert v\left(r-\rho \right)\Vert }^2\right)\text{d}r}\\ \le C\left(\omega \right)\left(1+R\left(\tau ,\omega \right)\right)(R\left(\tau ,\omega \right)+M(\; \tau \; ))\end{array}$

证毕。

引理3 若假设H₁，H₂成立，则有

$\begin{array}{c}{\displaystyle {\int }_{\tau -\rho }^{\tau }{\Vert \frac{\partial v}{\partial r}\Vert }^2\text{d}r}\le C\left(\omega \right){\displaystyle {\int }_{\tau -\rho }^{\tau }\left({\Vert D^{4}v\left(r\right)\Vert }^2+{\Vert D^{2}v\left(r\right)\Vert }^2+{\Vert f\left(r\right)\Vert }^2+c{\Vert v\left(r\right)\Vert }^2+{\Vert v\Vert }^2{\Vert Dv\Vert }^2+{\Vert v\left(r-\rho \right)\Vert }^2\right)\text{d}r}\\ \le c\left(\omega \right)\left(1+R\left(\tau ,\omega \right)\right)(R\left(\tau ,\omega \right)+M(\; \tau \; ))\end{array}$

其中 $R\left(\tau ,\omega \right)$ 与 $M\left(\tau \right)$ 由(3)定义。

证明：将(2)式与 $\frac{\partial }{\partial r}v\left(r,\tau -t,{\theta }_{-\tau }\omega ,\varphi \right)$ 在空间 $L^2\left(I\right)$ 上作内积，有

$\begin{array}{l}{\Vert \frac{\partial v}{\partial r}\Vert }^2+\alpha \left(D^{4}v,\frac{\partial v}{\partial r}\right)+\left(D^{2}v,\frac{\partial v}{\partial r}\right)+{\text{e}}^{z\left({\theta }_{r-\tau }\omega \right)}\left(vDv,\frac{\partial v}{\partial r}\right)\\ ={\text{e}}^{-z\left({\theta }_t\omega \right)}\left(f\left(r\right),\frac{\partial v}{\partial r}\right)+{\text{e}}^{-z\left({\theta }_t\omega \right)}\left(g\left({\text{e}}^{z\left({\theta }_{t-\rho }\omega \right)}v\left(t-\rho \right)\right),\frac{\partial v}{\partial r}\right)+z\left({\theta }_t\omega \right)\left(v,\frac{\partial v}{\partial r}\right)\end{array}$

由Young不等式

$\begin{array}{l}-\gamma \left(D^{4}v,\frac{\partial v}{\partial r}\right)-\left(D^{2}v,\frac{\partial v}{\partial r}\right)+{\text{e}}^{-z\left({\theta }_{r-\tau }\omega \right)}\left(f\left(r\right),\frac{\partial v}{\partial r}\right)+z\left({\theta }_t\omega \right)\left(v,\frac{\partial v}{\partial r}\right)\\ \le \frac{1}{6}{\Vert \frac{\partial v}{\partial r}\Vert }^2+c{\Vert D^{4}v\left(r\right)\Vert }^2+c{\Vert D^{2}v\left(r\right)\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2+c{\Vert z\left({\theta }_{r-\tau }\omega \right)v\left(r\right)\Vert }^2\end{array}$

由Agmon不等式

${\text{e}}^{z\left({\theta }_{r-\tau }\omega \right)}\left(vDv,\frac{\partial v}{\partial r}\right)\le \frac{1}{6}{\Vert \frac{\partial v}{\partial r}\Vert }^2+c{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}{\Vert v\Vert }^2{\Vert D^{2}v\Vert }^2$

由假设H₂

$\begin{array}{l}{\text{e}}^{-z\left({\theta }_t\omega \right)}\left(g\left({\text{e}}^{z\left({\theta }_{t-\rho }\omega \right)}v\left(t-\rho \right)\right),\frac{\partial v}{\partial r}\right)\\ \le \frac{1}{6}{\Vert \frac{\partial v}{\partial r}\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert g\left({\text{e}}^{z\left({\theta }_{r-\tau -\rho }\omega \right)}v\left(r-\rho \right)\right)\Vert }^2\\ \le \frac{1}{6}{\Vert \frac{\partial v}{\partial r}\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\text{e}}^{2z\left({\theta }_{r-\rho -\tau }\omega \right)}{\Vert v\left(r-\rho \right)\Vert }^2\end{array}$

从而

$\begin{array}{c}{\Vert \frac{\partial v}{\partial r}\Vert }^2\le c({\Vert D^{4}v\left(r\right)\Vert }^2+{\Vert D^{2}v\left(r\right)\Vert }^2+{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\Vert f\left(r\right)\Vert }^2+c{\Vert z\left({\theta }_{r-\tau }\omega \right)v\left(r\right)\Vert }^2\\ +c{\text{e}}^{2z\left({\theta }_{r-\tau }\omega \right)}{\Vert v\Vert }^2{\Vert Dv\Vert }^2+c{\text{e}}^{-2z\left({\theta }_{r-\tau }\omega \right)}{\text{e}}^{2z\left({\theta }_{r-\rho -\tau }\omega \right)}{\Vert v\left(r-\rho \right)\Vert }^2)\end{array}$

积分

$\begin{array}{c}{\displaystyle {\int }_{\tau -\rho }^{\tau }{\Vert \frac{\partial v}{\partial r}\Vert }^2\text{d}r}\le C\left(\omega \right){\displaystyle {\int }_{\tau -\rho }^{\tau }\left({\Vert D^{4}v\left(r\right)\Vert }^2+{\Vert D^{2}v\left(r\right)\Vert }^2+{\Vert f\left(r\right)\Vert }^2+c{\Vert v\left(r\right)\Vert }^2+{\Vert v\Vert }^2{\Vert Dv\Vert }^2+{\Vert v\left(r-\rho \right)\Vert }^2\right)\text{d}r}\\ \le C\left(\omega \right)\left(1+R\left(\tau ,\omega \right)\right)(R\left(\tau ,\omega \right)+M(\; \tau \; ))\end{array}$