【作者】 李勇；

【导师】 李扬荣；

【作者基本信息】 西南大学 ， 概率论与数理统计， 2021， 硕士

【摘要】 本文主要研究具有周期初值边界条件的随机时滞广义Kuramoto-Sivashinsky(KS)方程拉回吸引子的存在性唯一性及其当时滞时间趋于零时该吸引子的上半连续性.具体方程如下:这里τ ∈ R,α>0,β>0,γ>0,常数ρ>0是系统的时滞时间,h是缓增的时间依赖的外力项,f,φ,g是非线性项,F是带有时滞的非线性项.第一部分介绍了本文的研究背景及其现状.第二部分引入了动力系统的拉回随机吸引子的相关概念和存在唯一性定理.第三部分证明了带有周期性初值条件的非自治随机时滞广义KS方程拉回随机吸引子的存在性.为了得到拉回随机吸引子的存在性,系统在状态空间上的渐近紧性和闭可测拉回随机吸收集的存在性都需要得到证明.这里的难点在于系统在状态空间上的渐近紧性,我们需要利用广义KS方程解的高阶导数的估计以及Arzela-Ascoli定理去得到系统在状态空间上的渐近紧性.最后一部分证明时滞时间趋于零时拉回随机吸引子的上半连续性.时滞时间为任意值时第三部分的结论都成立,其结论就不需要重复证明.因此,本部分的难点为证明时滞时间趋于零时系统的稳定性,这需要将第三部分中的一些结论进行转化来运用到其中.更多还原

【Abstract】 This paper mainly studies the existence and uniqueness of the pullback attrac-tor of the stochastic delay generalized Kuramoto-Sivashinsky equation with periodic initial value boundary conditions.Moreover,the upper semicontinuity of the pull-back attractor when the time delay tends to zero is established.The specific equation is as follows:where τ∈R,α>0,β>0,γ>0,constant ρ>0 is the delay time of the system,h is a tempered time-dependent forcing,f,φ,g are nonlinear terms,and F is a nonlinear term with delay.The first part introduces the research background and current situation of this article.The second part introduces the related concepts and the existence and unique-ness theorem of the pullback random attractor of the dynamical system.The third part proves the existence of the pullback random attractor of the nonautonomous stochastic delay generalized Kuramoto-Sivashinsky equation with periodic initial value conditions.In order to obtain the existence of the pullback random attractor,Both the asymptotically compactness of the dynamical system in the state space and the existence of the closed measurable pullback absorbing set need to be proved.The difficulty here is mainly the asymptotically compactness of the dynamical system in the state space,we need to use the estimation of the solution of the higher order derivative of the generalized Kuramoto-Sivashinsky equation and the Arzela-Ascoli theorem to obtain the asymptotically compactness of the dynamical system in the state space.The last part proves the upper semicontinuity of the pullback random attractor when the delay time tends to zero.The conclusions of the third part are correct when the delay time is any value,and the conclusions do not need to be proved repeatedly.Therefore,the difficulty in this part is to prove the stability of the dynamical system when the delay time tends to zero.This requires transforming some of the conclusions in the third part to apply them.更多还原