两类非自治扩散方程动力学行为的研究

发布时间：2018-06-16 13:59

  本文选题：非自治扩散方程 + 动力边界　； 参考：《兰州大学》2016年博士论文

【摘要】：这篇博士论文我们研究非自治动力边界扩散方程和非自治分数次扩散方程解的长时间行为,分别建立新的先验估计,得到一系列新而且深刻的结果.全文分六章.第一章,我们简要回顾了动力系统的发展现状和拉回吸引子的提出背景,分析了带动力边界非自治扩散方程和非自治分数次扩散方程的意义、研究进展和思想.第二章,给出本文要用到的基础知识.第三章,研究带动力边界非自治反应扩散方程的动力学行为.首先建立了这类方程高阶可积性的先验估计定理.其次证明对任意δ0,方程的(L~2×L~2,L~2×L~2)拉回?-吸引子按L~(2+δ)×L~(2+δ)-范数拉回吸引每个L~2×L~2-有界集.最后证明了(L~2×L~2,L~2×L~2)拉回?-吸引子能按H1×H12范数拉回吸引每个L~2×L~2-有界集.第四章,研究带动力边界非自治p-Laplacian方程的动力学行为.首先建立高阶可积性的先验估计定理.其次证明对任意δ0,方程的(L~2×L~2,L~2×L~2)拉回?-吸引子按L~(2+δ)×L~(2+δ)-范数拉回吸引每个L~2×L~2-有界集.第五章,研究全空间上非自治分数次反应扩散方程的动力学行为.首先回顾分数次Sobolev空间的基本内容,并证明一个基本结论,参见引理5.1.8.其次证明方程存在(L~2,L~2)上的拉回?_μ-吸引子.再次,建立了这类方程高阶可积性的先验估计定理,参见定理5.2.1.基于这个结论,我们证明了对任意δ∈[0,∞),(L~2,L~2)拉回?_μ-吸引子能按L~(2+δ)-范数拉回吸引每个L~2-有界集.最后证明L~2拉回?_μ-吸引子能按H~s-范数拉回吸引每个L~2-有界集.特别地,证明了存在H~s中的拉回?_μ-吸引子.第六章,基于所取得的研究成果,列出部分将进一步研究的问题。
[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，

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