| One-dimensional damped semilinear wave equations is considered in this paper. with Dirichlet boundary condition u(0)=u(L)=0, and initial conditions u(x,0)=u0(x), ut(x,0)=u1(x). WhereΩ=(0, L),α>0, f∈L2(Ω).In studying of infinite-dimensional dynamical systems, Sine-Gordon equation is an important model and have important applications in many researching yields. In this paper, Sine-Gordon equation with damped is considered.Because Sine-Gordon equation is a wave equation with dissipative, we should retain dissipative property of equation possibly when we structure different scheme. In studying of infinite-dimensional dynamical systems,global attractor is an important notion for describing long-time behavior of dynamical systems. The finally state of dynamical systems is decided by global attractor. So, it is extremely important to study the existence of global attractor when studying long-time behavior of dynamical systems.In the first chapter of this paper, background of Sine-Gordon equation and domestic and international research results are introduced, meanwhile a one-dimensional damped semilinear wave equation (including the Sine-Gordon equation with damped) is reviewed. In the second chapter, some notations, notions and lemmas which can be used following are introduced. In the third chapter, we direct at the character of Sine-Gordon equation with damped structuring fully different scheme firstly; Then, the existence and unique of solution of fully different scheme are proved by Leray-Schauder fix point theorem. The next,for solution of different scheme, we get a priori estimate about time variant. Finally, based above researching we get stability convergence and error estimate of fully discrete different scheme. In the forth chapter, we discuss dynamical property of discrete dynamical systems which were generated by fully discrete different scheme with damped. The existence of global attractor is proved for the discrete dynamical systems. |
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