Semi-uniform Compact Attractors Of Non-autonomous Stochastic Sine-Gordon Equation

This thesis studies the semi-uniform compactness of pullback attractors for non-autonomous dynamical systems.We focus on the existence of backward compact and forward compact attractors for the non-autonomous stochastic sine-Gordon e-quation.Firstly,we introduce the background of non-autonomous stochastic dynamical systems and non-autonomous random attractors,and the background and research status of non-autonomous stochastic sine-Gordon equation.Secondly,we mainly establish an important criterion for the backward compact-ness of non-autonomous random attractors,that is,a backward compact random attractor exists if a non-autonomous random dynamical system is bounded dissipa-tive and backward asymptotically compact.We also obtain both backward compact and periodic random attractor from a periodic and locally asymptotically compact system.As an application,we consider the following non-autonomous stochastic sine-Gordon equation with multiplicative noise on the bounded domain Q:where ?>O,? ? R and Q is a bounded domain in Rn with arbitrary dimensions.If we assume that the density of noise is small enough and the force is backward tempered and backward complement-small,then we obtain a backward compact random attractor on the universe consisted of all backward tempered sets.Also,we obtain both backward compactness and periodicity of the attractor under the assumption of a periodic force.At last,we show the new concept of a forward controller.A pullback random attractor is called forward controllable if its time-component is semi-continuous to a compact set in the future,and the minimum among all such compact limit-sets is called a forward controller.The existence of a forward controller closely relates to the forward compactness of the attractor,which is further argued by the forward-pullback asymptotic compactness of the system.We also establish the theoretical result of the existence of a forward compact random attractor,that is,if the non-autonomous random dynamical system ? is a forward-pullback compactness and exists a decreasing absorption set,then ? has a unique forward compact attractor,which leads to the existence of a forward controller.As an application,we study the non-autonomous stochastic sine-Gordon equation with multiplicative noise on the unbounded domain Rn:where W is a two-sided scalar Wiener process and the multiplicative noise is in the sense of Stratonovich integrals.The measurability of the attractor is proved by considering two different universes and showing their equal.