Existence And Hausdorff Dimension Of The Random Attractor For The Strongly Damped Sine-Gordon Equation With White Noise

The random attractor is that of central parts of the asymptotic dynamics of the stochastic differential equation.This paper is devoted to the existence of the random attractor and the estimate of the upper bound of the Hausdorff dimension of the random attractor for the strongly damped stochastic sine-Gordon equation with important physical significance.In chapter 1,this paper introduce the development survey of the stochastic dynamical sys-tems and preliminary results and definitions(includes the definition of the stochastic,the existence of the random attractor and the estimate of the upper bound of the Hausdorff dimension of the random attractor), the basic function spaces and frequently used inequalities such as Young's Inequality, Gronwall's Inequality and Holder's Inequality, then, the author briefly introduce the research work of this paper.The research works of this paper consist of two chapter.In chapter 2, the author consider the strongly damped stochastic sine-Gordon equation under the Dirichlet boundary condition.Firstly,the author prove that strongly damped stochastic sine-Gordon equation can generate a stochastic dynamical system.then,by introducing weight norm and splitting positivity of the linear operator in the corresponding evolution equation of the first order with respect to time, existence of a compact random attractor is shown which attracts all tempered random set.In chapter 3,the author obtain an estimate of the upper bound of the Hausdorff dimension of the random attractor attractting all tempered random set for the strongly damped stochastic sine-Gordon equation.The obtained upper bound of Hausdorff dimension decease as the strongly damping grows and is uniformly bounded for large strongly damping. Under certain condi-tions,the dimension is zero.In particular,the author point out that the upper bound of Hausdorff dimension of the random attractor is also the one of Hausdorff dimension of the global atractor for the corresponding to the deterministic system without noise.