关于几类非自治梁方程（组）解的长时间动力行为的研究
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摘要
无穷维动力系统作为非线性科学的一个主要的研究对象,其理论与方法在许多重要领域和众多学科中有着广泛的应用,并且有着悠久的研究历史.近年来,非自治梁方程(组)作为无穷维动力系统的中心内容之一,受到了数学及其它自然科学工作者的高度重视,更在生物、化学、流体力学等领域结出了丰硕的成果.
本学位论文主要研究了非自治梁方程(组)系统的最终归宿,即：解的长时间动力学行为.由于吸引子是描述t→∞时系统的长时间动力学行为的重要指标,因此,它成为了无穷维动力系统研究的重点课题.这里,我们考虑的问题是：当t→∞时系统的相空间的任何相轨道是否从已知的初始状态出发又回到了原来的初始状态,以及是否被吸引到一个维数比原始空间更低的吸引子上?
对于非自治梁方程(组)所对应的无穷维动力系统的一致吸引子的存在性研究是本文的主要研究内容.文中,我们考虑了几类具有非线性阻尼系数的Kirchhoff型结构阻尼项、具有衰退记忆项、具有非线性阻尼项等一系列非自治梁方程(组)的一致吸引子的存在性问题.首先,我们将自治系统中的算子半群理论推广到非自治系统的过程理论,利用算子半群理论证明了系统存在连续解.其次通过能量的一致先验估计,构造了连续过程紧的或一致渐近紧的吸收集.最后通过过程分解技术,当外力项与时间相关时,将非自治系统所决定的过程{U(t,τ)}分解成两个小部分,并验证了一个满足压缩性质,另一个满足紧致性质.从而获得了由非自治系统所生成的过程存在一致吸引子.
本文共分六章,具体内容如下.
第一章,在阐述动力系统、无穷维动力系统和吸引子的应用背景的同时,介绍了吸引子的存在性的基本理论,以及自治和非自治系统的区别及其研究的进展概况.此外,还简单地介绍了本文所讨论的主要研究问题.
第二章,简要列举了本文用到的一些基本概念及理论.
第三章,在材料的粘性效应和非线性外阻尼作用下,考虑了较一般的具有非线性阻尼系数的带Kirchhoff型结构阻尼项的非自治梁方程在齐次Dirichlet边界条件下,当外力项与时间相关时,获得由非自治系统所生成的过程在空间H02(Ω)×L2(Ω)中一致吸引子的存在性.
第四章,当非线性项满足临界的Sobolev指数增长条件时,考虑了非自治情形下,具有衰退记忆项的非经典双曲梁方程当外力项h(x,t)依赖于时间并且为平移有界,而不是平移紧函数的时候,通过渐近非自治偏微分方程的极限集的性质,证明了在适当的参数范围内,对应于非自治系统所生成的过程族{Uh(t,τ),t≥τ,τ∈R}在弱拓扑空间H02(Ω)×L2(Ω)×Lμ2(R+;H02(Ω))和强拓扑空间D(A)×H02(Ω)×Lμ2(R+;D(A))中存在一致吸引子.
第五章,在非线性阻尼和热效应作用下,讨论了带强阻尼项的粘弹性非自治热弹耦合梁方程组当非自治外力项与时间相关,并且是平移紧的时侯,我们证明了系统所生成的解过程在空间H02(Ω)×L2(Ω)×L2(Ω)中存在一致吸引子.同时,我们还发现对于一定的参数范围内的吸引子,其结构是非常简单的,即：吸引子指数地吸引方程组的其它解,是方程组的有界完全轨道的一切值的唯一闭包.
第六章,在齐次Dirichlet边界条件下,研究了具有线性记忆项的非经典的非自治耦合梁方程组证明了当非线性项满足临界指数增长,且对于任意的非自治外力项是平移有界而非平移紧时,方程组具有一致吸引子,即：周期解唯一的指数吸引任何有界集.
Infinite dimensional dynamical system is an important research direction of nonlinear science. It has a long history, and the theory and method of it has comprehensive applications in many important areas and in many disciplines. In recent years, non-autonomous beam equations as one of central issues of infinite dimensional dynamical system, was attached great importance to mathematics and other natural science worker. And, it has made great achievements in biology, chemistry, fluid mechanics, etc.
The thesis is devoted to the study of non-autonomous beam equations--the long-time dynamical behavior of solutions. Attractor is an important indicator of the long-time dynamical behavior as time tends to infinity. Thus, it has become the key topics in the study of infinite dimensional dynamical system. Now, the question to consider is if the any orbit of phase space is from the known initial state back to the original on the initial state, and, if is attracted to a dimension space more than the original low attractor as time tends to infinity?
The study of the existence of uniform attractors for the infinite dimensional dynamical system of non-autonomous beam equations has become a main topic. In this thesis, we considered several cases about the existence of uniform attractors, such as the nonlinear viscoelastic beam equation with the Kirchhoff structural damped terms, the beam equation with linear memory type, the strongly damped nonlinear beam equations, etc. Firstly, we will promote the semigroup theorem of autonomous system to the process theorem of non-autonomous system and prove the existence of continuous solution by the operator semigroup theorem. Then, through a prior estimate of the energy, we structure the compact and uniformly asymptotically compact absorbing set of continuous process. Finally, using the process decomposition technique, we will decompose the corresponding family of processes {U(t,τ)} into two fractions, which one satisfies the squeezing property and the other is uniformly compact. Furthermore, we obtain the existence of uniform attractors of the process generated by non-autonomous system.
The thesis consists of six chapters, and the detailed content included following aspects.
In Chapter1, we not only exposit the background of dynamics, infinite dimensional dynamical system and attractors in the literature, but also introduce the basic theorem on the existence of attractors. Meanwhile, we illustrate the difference and its research progress of autonomous and non-autonomous system. In addition, we simply introduced the main research problems discussed in this thesis.
In Chapter2, some basic concept and theory that we will use in the thesis are presented.
Chapter3is concerned with the general Kirchhoff type non-autonomous beam equation with a nonlinear structural damped coefficient, under the material viscosity effect and the nonlinear damping effect. And under the homogeneous boundary condition, when the external force is time-dependent, the existence of uniform attractors of the process determined by non-autonomous system is obtained in the space H02(Ω)×L2(Ω).
In Chapter4, when the nonlinearity satisfies critical Sobolev exponential growth condition, the non-classic hyperbolical beam equation with fading memory in the case of non-autonomous is discussed. when the forcing term only translation bounded, not translation compact, we prove the existence of uniform attractors of the corresponding family of processes {Uh(t,τ),t≥τ}in weak topological space H02(Ω)×L2(Ω)×Lμ2(R+;H02(Ω)) and in strong topological space D(A)×H02(Ω)×Lμ2(R+;D(A))in a certain parameter region, through the properties of limit set of the asymptotic non-autonomous partial differential equations.
Chapter5discusses the non-autonomous viscoelastic coupled beam equations with strongly damped term under the nonlinear damping and the thermal effect, If the time-dependent forcing term is translation compact, the uniform attractor of solution process is obtained, and we can find that it has a simple structure. That is to say, the attractor attracts all the other solutions exponentially, and it is the only closure of all the solutions of the bounded completely orbit of the equations.
In Chapter6, under the homogeneous Dirichlet boundary condition, we study the non-autonomous non-classical coupled beam equations with linear memory, And, when the nonlinear term satisfies critical exponential growth, we prove the equations possess the uniform attractor. Namely, the periodic solution attracts any bounded set exponentially. Here, the any forcing term is translation bounded, not translation compact.
本学位论文主要研究了非自治梁方程(组)系统的最终归宿,即：解的长时间动力学行为.由于吸引子是描述t→∞时系统的长时间动力学行为的重要指标,因此,它成为了无穷维动力系统研究的重点课题.这里,我们考虑的问题是：当t→∞时系统的相空间的任何相轨道是否从已知的初始状态出发又回到了原来的初始状态,以及是否被吸引到一个维数比原始空间更低的吸引子上?
对于非自治梁方程(组)所对应的无穷维动力系统的一致吸引子的存在性研究是本文的主要研究内容.文中,我们考虑了几类具有非线性阻尼系数的Kirchhoff型结构阻尼项、具有衰退记忆项、具有非线性阻尼项等一系列非自治梁方程(组)的一致吸引子的存在性问题.首先,我们将自治系统中的算子半群理论推广到非自治系统的过程理论,利用算子半群理论证明了系统存在连续解.其次通过能量的一致先验估计,构造了连续过程紧的或一致渐近紧的吸收集.最后通过过程分解技术,当外力项与时间相关时,将非自治系统所决定的过程{U(t,τ)}分解成两个小部分,并验证了一个满足压缩性质,另一个满足紧致性质.从而获得了由非自治系统所生成的过程存在一致吸引子.
本文共分六章,具体内容如下.
第一章,在阐述动力系统、无穷维动力系统和吸引子的应用背景的同时,介绍了吸引子的存在性的基本理论,以及自治和非自治系统的区别及其研究的进展概况.此外,还简单地介绍了本文所讨论的主要研究问题.
第二章,简要列举了本文用到的一些基本概念及理论.
第三章,在材料的粘性效应和非线性外阻尼作用下,考虑了较一般的具有非线性阻尼系数的带Kirchhoff型结构阻尼项的非自治梁方程在齐次Dirichlet边界条件下,当外力项与时间相关时,获得由非自治系统所生成的过程在空间H02(Ω)×L2(Ω)中一致吸引子的存在性.
第四章,当非线性项满足临界的Sobolev指数增长条件时,考虑了非自治情形下,具有衰退记忆项的非经典双曲梁方程当外力项h(x,t)依赖于时间并且为平移有界,而不是平移紧函数的时候,通过渐近非自治偏微分方程的极限集的性质,证明了在适当的参数范围内,对应于非自治系统所生成的过程族{Uh(t,τ),t≥τ,τ∈R}在弱拓扑空间H02(Ω)×L2(Ω)×Lμ2(R+;H02(Ω))和强拓扑空间D(A)×H02(Ω)×Lμ2(R+;D(A))中存在一致吸引子.
第五章,在非线性阻尼和热效应作用下,讨论了带强阻尼项的粘弹性非自治热弹耦合梁方程组当非自治外力项与时间相关,并且是平移紧的时侯,我们证明了系统所生成的解过程在空间H02(Ω)×L2(Ω)×L2(Ω)中存在一致吸引子.同时,我们还发现对于一定的参数范围内的吸引子,其结构是非常简单的,即：吸引子指数地吸引方程组的其它解,是方程组的有界完全轨道的一切值的唯一闭包.
第六章,在齐次Dirichlet边界条件下,研究了具有线性记忆项的非经典的非自治耦合梁方程组证明了当非线性项满足临界指数增长,且对于任意的非自治外力项是平移有界而非平移紧时,方程组具有一致吸引子,即：周期解唯一的指数吸引任何有界集.
Infinite dimensional dynamical system is an important research direction of nonlinear science. It has a long history, and the theory and method of it has comprehensive applications in many important areas and in many disciplines. In recent years, non-autonomous beam equations as one of central issues of infinite dimensional dynamical system, was attached great importance to mathematics and other natural science worker. And, it has made great achievements in biology, chemistry, fluid mechanics, etc.
The thesis is devoted to the study of non-autonomous beam equations--the long-time dynamical behavior of solutions. Attractor is an important indicator of the long-time dynamical behavior as time tends to infinity. Thus, it has become the key topics in the study of infinite dimensional dynamical system. Now, the question to consider is if the any orbit of phase space is from the known initial state back to the original on the initial state, and, if is attracted to a dimension space more than the original low attractor as time tends to infinity?
The study of the existence of uniform attractors for the infinite dimensional dynamical system of non-autonomous beam equations has become a main topic. In this thesis, we considered several cases about the existence of uniform attractors, such as the nonlinear viscoelastic beam equation with the Kirchhoff structural damped terms, the beam equation with linear memory type, the strongly damped nonlinear beam equations, etc. Firstly, we will promote the semigroup theorem of autonomous system to the process theorem of non-autonomous system and prove the existence of continuous solution by the operator semigroup theorem. Then, through a prior estimate of the energy, we structure the compact and uniformly asymptotically compact absorbing set of continuous process. Finally, using the process decomposition technique, we will decompose the corresponding family of processes {U(t,τ)} into two fractions, which one satisfies the squeezing property and the other is uniformly compact. Furthermore, we obtain the existence of uniform attractors of the process generated by non-autonomous system.
The thesis consists of six chapters, and the detailed content included following aspects.
In Chapter1, we not only exposit the background of dynamics, infinite dimensional dynamical system and attractors in the literature, but also introduce the basic theorem on the existence of attractors. Meanwhile, we illustrate the difference and its research progress of autonomous and non-autonomous system. In addition, we simply introduced the main research problems discussed in this thesis.
In Chapter2, some basic concept and theory that we will use in the thesis are presented.
Chapter3is concerned with the general Kirchhoff type non-autonomous beam equation with a nonlinear structural damped coefficient, under the material viscosity effect and the nonlinear damping effect. And under the homogeneous boundary condition, when the external force is time-dependent, the existence of uniform attractors of the process determined by non-autonomous system is obtained in the space H02(Ω)×L2(Ω).
In Chapter4, when the nonlinearity satisfies critical Sobolev exponential growth condition, the non-classic hyperbolical beam equation with fading memory in the case of non-autonomous is discussed. when the forcing term only translation bounded, not translation compact, we prove the existence of uniform attractors of the corresponding family of processes {Uh(t,τ),t≥τ}in weak topological space H02(Ω)×L2(Ω)×Lμ2(R+;H02(Ω)) and in strong topological space D(A)×H02(Ω)×Lμ2(R+;D(A))in a certain parameter region, through the properties of limit set of the asymptotic non-autonomous partial differential equations.
Chapter5discusses the non-autonomous viscoelastic coupled beam equations with strongly damped term under the nonlinear damping and the thermal effect, If the time-dependent forcing term is translation compact, the uniform attractor of solution process is obtained, and we can find that it has a simple structure. That is to say, the attractor attracts all the other solutions exponentially, and it is the only closure of all the solutions of the bounded completely orbit of the equations.
In Chapter6, under the homogeneous Dirichlet boundary condition, we study the non-autonomous non-classical coupled beam equations with linear memory, And, when the nonlinear term satisfies critical exponential growth, we prove the equations possess the uniform attractor. Namely, the periodic solution attracts any bounded set exponentially. Here, the any forcing term is translation bounded, not translation compact.
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