Study On Fractal Dimensions Of Random Attractors For Stochastic F-N Equations And Stochastic BBM Equation

This paper considers the asymptotic behavior of solutions for non-autonomous FitzHugh-Nagumo(F-N)system and Benjamin-Bona-Mahony(BBM)system driv-en by additive white noise.We prove the existence of random attractors and the finiteness of its fractal dimension.Random attractor is the tool method to depict the stochastic dynamical system.In the first chapter,we mainly introduce the status of stochastic dynamical system and major ways.At the same time,some basic concepts,some classical in-equalities will be used in this paper.In the second chapter,we consider the asymptotic behavior of solutions for non-autonomous F-N equation.Firstly,we apply the method of compact imbedding to prove the existence of fractal dimension of the random attractor for stochastic F-N system in this chapter.Secondly,we use the method of estimating an upper bound of fractal dimension of a random invariant set of a random dynamical sys-tem,and we calculate expectation of some random variables to prove the bound-edness of fractal dimension of the random attractors for this system.In the third chapter,we investigate the existence of random attractor and the finiteness of its fractal dimension for stochastic BBM equation driven by additive white noise on unbounded domains.In this chapter,the main method is uniform estimates to prove the existence of random attractor for this equation.Then,we present criterion for estimating an upper bound of the fractal dimension of a ran-dom invariant set of a random dynamical system on a separable Banach space.And on this basis we apply expectation of random variables to prove the bound-edness of fractal dimension of random attractors for this equation.In the forth chapter,we summarize the conclusions obtained in this article and point out problem remaining to be solved.