The Global Attractor For The Nonlinear Elastic Rod Systems

The purpose of studying dynamical systems is to find out the law of natural phe-nomena with the time varying. Compared with the finite-dimensional dynamical systems, the infinite-dimensional dynamical systems are more general in mathematics and exist almost everywhere in physics. The infinite-dimensional dynamical systems study the spatial chaos which is a complicated and unreadable concept. Because of the absorbency and invariance, the global attractors are one of the best tools to describe the chaos and can perfectly describe the long time behavior of the systems.Viewed from the nonlinear damping and thermal effect, this paper is concerned with the existence of global attractors to two kinds of elastic rod systems. The first one is the coupled rod system with a strong damping and thermal effect whereα1,β1,α2 are positive constants.The second one is the Kirchhoff type equation with a nonlinear damping coeffic-ient whereαare positive constants.The paper is arranged as followsIn Chapterl, we introduce the background of elastic rod systems and current research situation, and give the main research results of this paper.In Chapter2, we give some definitions and lemmas relate to the paper, and make a simple introduction to Sobolev space.In Chapter3, we consider the initial boundary value problem for a class of coupled elastic rod equation with a strong damping and thermal effect. By using the theory of operator semigroup, we prove the existence and uniqueness of the solution for this system; We obtain the existence of the global attractor for this system by means of the classic decomposed idea of semigroup.In Chapter4, we discuss the initial boundary value problem for the Kirchhoff type elastic rod equation with a nonlinear damping coefficient. By Faedo-Galerkin method, we derive that this equation has a continuous, unique solution; The existence of the global attractor for this equation is proved through an important lemma in Chapter2.In Chapter5, the summary of this dissertation and prospects of researches are presented.