Wave Equation With Damping Term Finite Difference To The Long Time Behavior Of The Decomposition

Infinite dimensional dynamical system is an important branch of dynamical sys-tems, which is widely used in many areas, and gradually become the main study object and subjects of the dynamical system. In the study, where the global attrac-tor determines the final state of the infinite-dimensional dynamical systems, and reflecting the long time behavior of infinite dimensional dynamical system. With the rapid development of computer, numerical simulation by computers of in the infinite-dimensional dynamical system has become surprisingly mature, the finite difference method is a numerical method based on the discrete equations and solving region, it makes a continuous infinite dimensional dynamical system to be discrete,so that it can be numerical approximated by computers, and which is one of the most commonly used method. In this paper, we consider the problem of the long time behavior of the one-dimensional Damped semilinear wave equations whit the means of finite difference methods.Here gives the semilinear damped wave equation where(x, l)∈(?) x (0,+∞), and(?)=[0, L]. with the boundary condition and the initial condition is In this paper, the first chapter outlines the development history and current situ-ation of infinite dimensional dynamical system, as well as the application of finite difference method to solve the problem of dynamical system, and introduces the gradient dynamical systems; Related symbols and representation are defined and described in chapter II, also some important concepts and lemmas in this article arereviewed; Chapter III introduces the Lyapunov functional and gives priori estima-tion; Chapter IV analyzes the convergence and stability of the diference equations,also the error analysis; Chapter V researches the gradient dynamical system gen-erated by the semi-discrete diferential equations, and proves that the existence ofglobal attractor.