The Existence Of Generalized Sine-Gordon Equation Attractor

Sine-Gordon equation is a nolinear form of telegram equation,it is closely related with Klein-Gordon equation in relativistic field theory.This equation is recognized by people in nineteenth century.With the deep study, people pay more and more attention to this equation.It’s form is as follow.We will solve some problem.The existence of the equation’s solution,the existence of absorber set,the existence of compact attractor.In the paper we will use Faedo-Galerkin method to study the following sine-gordon equation in the Sobolev space. In the initial conditions And boundary conditionswhere, x∈[0,l],t∈[0,+∞), g(·) satisfied g(h)≤c(1+|h|3), u0, u1are given functions.α>0, α∈R, f, u0, u1staisfied f∈C([O,T];H), u0∈V, u1∈HThis article will be divided into the following four-part study:1. We will make some simple sum-up and comments of the development and research of dynamic system and Sine-Gordon equation with this paper.2. This article will give some important concepts and lemmas.3. By Faedo-Galerkin method, Sobolev space theory and some inequality theory,we prove the existence and uniqueness of global solution of the Sine-Gordon function under initial and the first bound condition4. By the priori estimates,we prove the existence of absorbing set of the Sine-Gordon function under initial and the first bound condition in different sobolev space. By using the property of operator semigroup,we wrote S(t) in the form S1(t) and S2(t) for the operator semigroup of the Sine-Gordon equation,where S1(t) being uniformly compact and S2(t) being satifisfied the squeezing property.Then we proved the existence of global attractor of the dynamical system from Sine-Gordon equation.