本文選題：非自治擴(kuò)散方程 + 動(dòng)力邊界��； 參考：《蘭州大學(xué)》2016年博士論文

【摘要】：這篇博士論文我們研究非自治動(dòng)力邊界擴(kuò)散方程和非自治分?jǐn)?shù)次擴(kuò)散方程解的長時(shí)間行為,分別建立新的先驗(yàn)估計(jì),得到一系列新而且深刻的結(jié)果.全文分六章.第一章,我們簡要回顧了動(dòng)力系統(tǒng)的發(fā)展現(xiàn)狀和拉回吸引子的提出背景,分析了帶動(dòng)力邊界非自治擴(kuò)散方程和非自治分?jǐn)?shù)次擴(kuò)散方程的意義、研究進(jìn)展和思想.第二章,給出本文要用到的基礎(chǔ)知識.第三章,研究帶動(dòng)力邊界非自治反應(yīng)擴(kuò)散方程的動(dòng)力學(xué)行為.首先建立了這類方程高階可積性的先驗(yàn)估計(jì)定理.其次證明對任意δ0,方程的(L~2×L~2,L~2×L~2)拉回?-吸引子按L~(2+δ)×L~(2+δ)-范數(shù)拉回吸引每個(gè)L~2×L~2-有界集.最后證明了(L~2×L~2,L~2×L~2)拉回?-吸引子能按H1×H12范數(shù)拉回吸引每個(gè)L~2×L~2-有界集.第四章,研究帶動(dòng)力邊界非自治p-Laplacian方程的動(dòng)力學(xué)行為.首先建立高階可積性的先驗(yàn)估計(jì)定理.其次證明對任意δ0,方程的(L~2×L~2,L~2×L~2)拉回?-吸引子按L~(2+δ)×L~(2+δ)-范數(shù)拉回吸引每個(gè)L~2×L~2-有界集.第五章,研究全空間上非自治分?jǐn)?shù)次反應(yīng)擴(kuò)散方程的動(dòng)力學(xué)行為.首先回顧分?jǐn)?shù)次Sobolev空間的基本內(nèi)容,并證明一個(gè)基本結(jié)論,參見引理5.1.8.其次證明方程存在(L~2,L~2)上的拉回?_μ-吸引子.再次,建立了這類方程高階可積性的先驗(yàn)估計(jì)定理,參見定理5.2.1.基于這個(gè)結(jié)論,我們證明了對任意δ∈[0,∞),(L~2,L~2)拉回?_μ-吸引子能按L~(2+δ)-范數(shù)拉回吸引每個(gè)L~2-有界集.最后證明L~2拉回?_μ-吸引子能按H~s-范數(shù)拉回吸引每個(gè)L~2-有界集.特別地,證明了存在H~s中的拉回?_μ-吸引子.第六章,基于所取得的研究成果,列出部分將進(jìn)一步研究的問題。
[Abstract]:In this doctoral thesis, we study the long term behavior of solutions of nonautonomous dynamic boundary diffusion equations and nonautonomous fractional diffusion equations, and establish new prior estimates respectively, and obtain a series of new and profound results. The full text is divided into six chapters. In chapter 1, we briefly review the development of dynamic systems and the background of pull back attractor, analyze the significance of nonautonomous diffusion equations with dynamic boundary and fractional diffusion equations, and research progress and ideas. In the second chapter, the basic knowledge to be used in this paper is given. In chapter 3, the dynamical behavior of nonautonomous reaction-diffusion equations with dynamic boundary is studied. A priori estimate theorem for the higher order integrability of this kind of equations is established. Secondly, it is proved that for any 未 _ 0, the L ~ (2 脳 L ~ (2) L ~ (+) ~ (2 脳 L ~ (2) ~ (2) ~ (2 未) 脳 L ~ (2) C ~ (2 未) -norm pull back to attract every L ~ (2 脳 L ~ (2) -L ~ (2) -bounded set of the equation. Finally, it is proved that the pull-back attractor can draw back every L ~ (2 脳 L ~ (1) ~ (2 脳 L ~ (2) -bounded set according to H _ (1 脳 H _ (12) norm. In chapter 4, the dynamical behavior of nonautonomous p-Laplacian equations with dynamic boundary is studied. First, a priori estimate theorem of higher order integrability is established. Secondly, it is proved that for any 未 _ 0, the L ~ (2 脳 L ~ (2) L ~ (+) ~ (2 脳 L ~ (2) ~ (2) ~ (2 未) 脳 L ~ (2) C ~ (2 未) -norm pull back to attract every L ~ (2 脳 L ~ (2) -L ~ (2) -bounded set of the equation. In chapter 5, the dynamical behavior of nonautonomous fractional reaction-diffusion equations on the whole space is studied. This paper first reviews the basic contents of fractional Sobolev spaces, and proves a basic conclusion, see Lemma 5.1.8. Secondly, it is proved that there exists a pullback _ 渭 -attractor on the equation. Thirdly, a priori estimate theorem for the higher order integrability of this kind of equation is established, see Theorem 5.2.1. On the basis of this conclusion, we prove that for any 未 鈭，

本文編號：2026937