【摘要】：本文主要研究幾類帶衰退記憶的非線性發(fā)展方程的長時(shí)間行為.第一章主要介紹研究整體吸引子的一些方法以及某些帶衰退記憶的非線性偏微分方程的研究現(xiàn)狀,給出了本文的主要研究內(nèi)容和目的.第二章主要研究帶衰退記憶和臨界非線性的四階擬拋物方程的長時(shí)間行為.在過去歷史框架下,利用分解技巧和緊性轉(zhuǎn)移定理證明了對(duì)應(yīng)的動(dòng)力系統(tǒng)的整體吸引子存在性.第三章考慮帶非局部擴(kuò)散的非自治非經(jīng)典擴(kuò)散方程的拉回吸引子存在性.在適當(dāng)?shù)募僭O(shè)下,利用能量方法證明了兩個(gè)不同框架下相對(duì)應(yīng)的過程的極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個(gè)溫和條件下給定的全域上的拉回吸引子之間的關(guān)系.第四章處理非線性粘彈性Kirchhoff板方程的長時(shí)間動(dòng)力學(xué).通過對(duì)記憶核g和非線性項(xiàng)f附加一些增長性條件,證明了對(duì)應(yīng)的動(dòng)力系統(tǒng)的整體吸引子的存在性.此外,在次臨界情形時(shí),利用擬穩(wěn)定性性質(zhì)證明了這個(gè)吸引子具有有限Hausdorff和分形維數(shù).第五章考慮帶非線性阻尼的擬線性粘彈性方程的長時(shí)間行為.首先,運(yùn)用Galerkin方法證明了整體弱解的存在唯一性.其次,利用能量擾動(dòng)法得到了解能量的衰減估計(jì).最后,利用一個(gè)穩(wěn)定性不等式證明了整體吸引子的存在性。
[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.
【學(xué)位授予單位】：廣州大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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本文編號(hào)：2366503