Asymptotic Behavior Of Quintic Ginzburg-landau Equations Driven By Multiplicative Noise

In this Master Dissertation, the author mainly study the existence of the solution and the existence of the attractors of stochastic Ginzburg-Landau systems and get the results of the existence of the solution and the existence of the attractors. This paper is divided into five chapters:The first chapter is introduction and this chapter is divided into two parts. We introduce the research background, research status at home and abroad and what I have done, respectively.The second chapter is preliminary knowledge. In this chapter, we introduce basic symbols, function spaces, inequalities and some concepts, such as It? formula and the global attractor used in this paper.In Chapter 3 and chapter 4 we study the cubic-quintic stochastic Ginzburg-Landau equation and two coupled complex cubic-quintic stochastic Ginzburg-Landau equations, respectively.Due to the special linear multiplicative noise, the equation can be reduced to the equation with random coefficients by a suitable change of variables, so this problem can be solved pathwise by using some technologies. First, in view of the existence theorem of the solution of the deterministic equation, the existence of the solution to the equation with random coefficientsis can be obtained. Making use of the It? formula, Logarithmic form of unknown function can be obtained. Via the 2L-norm estimates of logarithmic form instead of the 2L-norm estimates of the unknown function, the stability of the solution of the random partial differential equation can be gained. Based on various norm estimates for the solution and the compact embedding theorem in Sobolev space, we can get the existence of random attractor of the equation.In fifth chapter, a summary of the author’s work in this paper along with an outlook for future research is given.