Long Time Behavior Of Random Long-Short Wave Equations

In this paper,the generalized(2+1)-dimensional non-autonomous long-short wave equations and the long time behavior of stochastic(2+1)-dimensional long-short wave equations were studied.We obtained the existence of uniform attractor and the approximate inertial manifolds for the generalized(2+1)-dimensional long-short wave equations.We proved the existence of the random attractor and stationary measure of the stochastic(2+1)-dimensional long-short wave equations.This article was divided into four parts.The first part,we introduced the physical background and theoretical knowledge for the infinite-dimensional dynamical systems,the stochastic differential equations and the long-short wave equations.We reviewed some research results and introduced the main research work of this paper.The second part,we considered the uniform attractor of generalized(2+1)-dimensional non-autonomous long-short wave equations.Firstly,using the priori estimate and the Gal?rkin method,the existence and uniqueness of the solution were obtained.Secondly,using the theory of non-autonomous systems,the existence of strong compact uniform attractor for the system was proved.The third part,we constructed the approximate inertial manifold of the generalized(2+1)-dimensional non-autonomous long-short wave equations.By using the method of expanding the phase plane and the projection operator,the approximate inertial manifold of this non-autonomous system was constructed.The fourth part,the existence of the random attractor and stationary measure of the stochastic(2+1)-dimensional long-short wave equations were proved.At first,we obtained the existence and uniqueness of the solution for the system by using It?formula,the prior estimate and the Gal?rkin method.Then we proved the existence of the random absorbing set and the random attractor for the equations.Finally,we considered the stationary measure of the equations.