U-Pullback Attractors For Stochastic Non-Autonmous Damped Sine-Gordon Equations

In this thesis,the asymptotic behavior of the non-autonomous stochastic cou-pled Sine-Gordon equations with additive noise is studied.We prove the existence and upper semi-continuity of attractor for stochastic non-autonmous damped Sine-Gordon equations.Sine-Gordon equation has many interesting physical phenomena and practical value,so it becomes one of an important model in the infinite dynamic system.Let ??Rn be a bounded open domain.We consider the following non-autonomous stochastic coupled Sine-Gordon equations:Where ui=ui(x,t),i=1,2 is defined the real value unknown function in ft x[r,+OO),r G M,a is the damping coefficient,and (?) is a time-independent function.And W(t) is the derivative form of the one-dimensional two-sided real-valued Wiener process (?) describes a white noise.The external force term depends on the time variable (?) and satisfy the following two conditions:(?)The literature[7,1178]studied the single Sine-Gordon equation.Furthermore,the existence of random attractors of the random Sine-Gordon equations has been studied,see for instance[17,19].The equations discussed in this paper generalize the corresponding equations in the literature[7]from three aspects:i.Two variables coupled to the Sine-Gordon equations;ii.Deterministic non-auionomous forcing;iii.Additive noise.This thesis contains four charpters:Chapter 1:The basic theory of non-autonomous stochastic dynamic systems and stochastic pullback attractors is introduced.Chapter 2:Using the O-U process,variable substitution 7 proves that the Sine-Gordon equations discussed have a unique solution and produce a continuous non-autonomous stochastic dynamic system(cocycle).Chapter 3:We derive uniform estimate of solutions to prove the existence of the U-absorbing set,and the decomposition technique of the solution proves that the dynamic system is asymptotically compact,which leads to the existence of attractor for stochastic non-autonmous damped Sine-Gordon equations.Chapter 4:By proving the convergence of the random dynamic system,it proves that the upper semi-continuity of the U-pullback attractor.