【摘要】：拉回吸引子是描述系統(tǒng)解的長(zhǎng)時(shí)間漸近行為的緊集,是研究無(wú)窮維動(dòng)力系統(tǒng)的重要工具.本文考慮無(wú)界區(qū)域上一類(lèi)波動(dòng)方程的拉回吸引子,獲得了無(wú)界區(qū)域上非自治Brinkman-Forchheimer方程和非自治Sine-Gordon方程拉回吸引子的存在性.全文共分為五章.第一章,介紹了無(wú)窮維動(dòng)力系統(tǒng)的背景,分自治和非自治兩種情形闡述了全局吸引子、一致吸引子和拉回吸引子的研究進(jìn)展情況,還介紹了本文的主要工作和思想.第二章,介紹了所要研究問(wèn)題的預(yù)備知識(shí),給出了文章需要用到的一些相關(guān)符號(hào)、不等式、定理以及相關(guān)理論,為接下來(lái)的討論做準(zhǔn)備.第三章,研究了無(wú)界區(qū)域L~2(Ω)上非自治Brinkman-Forchheimer方程拉回吸引子的存在性,針對(duì)無(wú)界區(qū)域上Sobolev嵌入不再是緊的,利用一致估計(jì)和截?cái)嗪瘮?shù)技巧證明了無(wú)界域上拉回吸引子的存在性,將無(wú)界區(qū)域分成兩部分,有界部分和無(wú)界部分,有界部分用Sobolev緊嵌入,無(wú)界部分則證明指數(shù)一致衰減到零,從而得到緊性.第四章,研究了無(wú)界區(qū)域H~1(R~n)×L~2(R~n)上非自治Sine-Gordon方程拉回吸引子的存在性,同樣的,此時(shí)Sobolev緊嵌入不再成立,先對(duì)方程的解進(jìn)行先驗(yàn)估計(jì),通過(guò)解的一致估計(jì)證明了方程拉回吸收集的存在性,而后構(gòu)造能量方程證明了解的拉回漸近緊性,從而得到Sine-Gordon方程在無(wú)界域上拉回吸引子的存在性.第五章,對(duì)本文的總結(jié)及對(duì)拉回吸引子研究前景的展望。
[Abstract]:The pull back attractor is a compact set which describes the long time asymptotic behavior of the solution of the system and is an important tool for the study of infinite dimensional dynamical systems. In this paper, we consider the pullback attractors of a class of wave equations in unbounded regions, and obtain the existence of pullback attractors for nonautonomous Brinkman-Forchheimer equations and nonautonomous Sine-Gordon equations in unbounded regions. The full text is divided into five chapters. In chapter 1, the background of infinite dimensional dynamical system is introduced. The research progress of global attractor, uniform attractor and pull back attractor is described in two cases of autonomy and nonautonomy. The main work and ideas of this paper are also introduced. In the second chapter, we introduce the preparatory knowledge of the problem, and give some relevant symbols, inequalities, theorems and relevant theories to be used in this paper, so as to prepare for the following discussion. In chapter 3, we study the existence of pullback attractors for the nonautonomous Brinkman-Forchheimer equations on the unbounded domain Ln 2 (惟). For the unbounded region, Sobolev embedding is no longer compact. By using uniform estimation and truncation function technique, the existence of pull back attractor on unbounded domain is proved. The unbounded region is divided into two parts, bounded part and unbounded part, bounded part is Sobolev compactly embedded, and unbounded part is proved that exponentially uniformly attenuates to zero. Thus the compactness is obtained. In chapter 4, we study the existence of the pull back attractor of the nonautonomous Sine-Gordon equation on the unbounded domain H1 (rn) 脳 Ln (rn). Similarly, the Sobolev compactness embedding is no longer true at this time, a priori estimate of the solution of the equation is made. The existence of the pull-back absorption set of the equation is proved by the uniform estimation of the solution, and the tension asymptotically compactness of the solution is proved by constructing the energy equation, and the existence of the trace-back attractor of the Sine-Gordon equation in the unbounded domain is obtained. The fifth chapter, the summary of this paper and the prospect of pull-back attractor research.
【學(xué)位授予單位】：廣州大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O175

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