Study On The Long-time Dynamical Behavior Of Solutions Of Several Classes Of Non-autonomous Beam Equation (Equations)

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.