几类带衰退记忆的非线性发展方程的长时间行为

发布时间：2018-12-06 18:59

【摘要】：本文主要研究几类带衰退记忆的非线性发展方程的长时间行为.第一章主要介绍研究整体吸引子的一些方法以及某些带衰退记忆的非线性偏微分方程的研究现状,给出了本文的主要研究内容和目的.第二章主要研究带衰退记忆和临界非线性的四阶拟抛物方程的长时间行为.在过去历史框架下,利用分解技巧和紧性转移定理证明了对应的动力系统的整体吸引子存在性.第三章考虑带非局部扩散的非自治非经典扩散方程的拉回吸引子存在性.在适当的假设下,利用能量方法证明了两个不同框架下相对应的过程的极小拉回吸引子的存在性.此外,建立了固定有界集的全域上的拉回吸引子和在一个温和条件下给定的全域上的拉回吸引子之间的关系.第四章处理非线性粘弹性Kirchhoff板方程的长时间动力学.通过对记忆核g和非线性项f附加一些增长性条件,证明了对应的动力系统的整体吸引子的存在性.此外,在次临界情形时,利用拟稳定性性质证明了这个吸引子具有有限Hausdorff和分形维数.第五章考虑带非线性阻尼的拟线性粘弹性方程的长时间行为.首先,运用Galerkin方法证明了整体弱解的存在唯一性.其次,利用能量扰动法得到了解能量的衰减估计.最后,利用一个稳定性不等式证明了整体吸引子的存在性。
[Abstract]:In this paper, we study the long-term behavior of several nonlinear evolution equations with recessionary memory. In the first chapter, we introduce some methods of studying global attractor and the present situation of some nonlinear partial differential equations with fading memory, and give the main contents and purposes of this paper. In chapter 2, the long time behavior of fourth order quasi parabolic equation with recessionary memory and critical nonlinearity is studied. In the framework of the past history, the existence of the global attractor of the corresponding dynamical system is proved by using the decomposition technique and the compactness transfer theorem. In chapter 3, we consider the existence of pull attractors for nonautonomous nonclassical diffusion equations with nonlocal diffusion. Under appropriate assumptions, the existence of minimal pull-back attractors in two different frames is proved by energy method. In addition, the relationship between the pull back attractor on a fixed bounded set and the pull back attractor on a given global domain under a mild condition is established. Chapter 4 deals with the long time dynamics of nonlinear viscoelastic Kirchhoff plate equations. By attaching some growth conditions to the memory kernel g and the nonlinear term f, the existence of the global attractor for the corresponding dynamical system is proved. In addition, in the subcritical case, it is proved that the attractor has finite Hausdorff and fractal dimensions by using the quasi-stability property. In chapter 5, the long time behavior of quasilinear viscoelastic equations with nonlinear damping is considered. Firstly, the existence and uniqueness of the global weak solution are proved by Galerkin method. Secondly, the energy decay estimation of the solution is obtained by using the energy perturbation method. Finally, the existence of global attractor is proved by using a stability inequality.
【学位授予单位】：广州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

【参考文献】