Random Attractors Of Sine-Gordon Equations With Additive Noise

In the paper, we discuss the existence of the random attractor of sine-Gordon equa-tions with additive noise. The sine-Gordon equation in the paper is a kind of nonlinear wave equation and one of the important models in infinite dimension dynamic system. The model is often used in Physics, such as used to describe the spread of dislocation grid work, the former resources driving Josephson junction dynamics, the general theory of elemental particle, the transmission of the expansion of the lipid membrane etc. Sine-Gordon equation is recognized in the19th century, and of great significance to quan-tum physics. With the deep understanding, this equation get more and more attention Sine-Gordon equation is a mathematical model which is used to describe the continu-ous Josephson junction[1]. In1962Josephson firstly used sine-Gordon type constitutive relationship in the Josephson junction for superconductors.The deterministic case of this kind of equation has been discussed by many scholars systematically e.g. Temma. Therefore, small irregularities have to be taken account. It is said, it is to research the random attractor of some equations after adding a random force which is a time white noise by many scholars for other equations.Attractor is one of the hot interesting problems in the world recently. In1994, Crauel. H. and Flandoli. F. gave the definition of the global attractor for random dynamic system by the definition of random set in the literature[2]. The global attractor is one of use-ful tools which has been introduced to describe the long-time behavior of the dynamical which is generated by partial differential equations. In the stochastic case, the global attractor which is complex is a compact random invariant set which attracts all deter-ministic bounded sets, and reflect the complexity of the dynamical systems at infinite distance.This article is devoted to the existence of the random attrator for damped sine-Gordon equations with homomgeneous Dirichlet boundary condition when there ar ran-dom terms. We consider the following equations: Let Ω be an open bounded set of R with a smooth boundary (?)Ω,u1t=(?)u1/(?)t, u2t=(?)u2/(?)t,u1=u1(x,t),u2=u2(x,t)are real-valued functions on Ω×[τ,+∞), for τ∈R,f1(x),f2(x)∈H2(Ω)∩H01(Ω) is time-independent function. W(t) is a one-dimensional two-side Wiener process on the complete probability space(θ,F,P), where θ={ω∈C(R, R):ω(0)=0}, F is the Borel sigma-algebra induced by the compact-open topology of Ω, P is a Wiener measure. Let {θt}t∈R:θ(·)=ω(·+t)-ω(t) is a family of measure preserving transformations on the space (θ,F,P). The random attractor and its property for damped sine-Gordon equations are instigated in the literature [2]. This paper will use the the development theory in the reference [3] to prove the existence of the random attrator for above equations.This paper is divided into four chapters:In Chapter one: This part mainly introduces the background and developing process of the knowledge related the article and gives us the basic knowledge and some symbols that will be used in this paper.In Chapter two: There is a unique solution of the above equations by using the appropriate transformation and method. Of course, the unique solution of equations(2.1.1) will generate a random RDS.In Chapter three: The random RDS has a random attractorIn Chapter four: More study about this equations are needed to be studied further.