Studies On The Long Time Behavior For Partly Dissipative Stochastic Systems

The long time behavior of the stochastic reaction diffusion equations with additive white noise and small diffusion coefficients and the modified SwiftHohenberg equation defined on unbounded domain driven by both deterministic non-autonomous and random external force were considered in this paper. By stochastic processes transformation and using the properties of the stochastic processes, we carry out the uniform priori estimates for the solution of the equations. We obtained existence of bounded absorbing sets, and through the estimation for the tail of the solution we proved the asymptotic compactness property of the solution operator for the consider equations. Finally, the existence of random attractors in the given space is proved. And the upper semi-continuity of random attractors is discussed. Summarizes the primary coverage which this article studies as follows to show:(1) In the first chapter, a simple overview of random dynamical systems is carried out. This section illustrates the present research situation and significance of random dynamical systems, and introduces some preparation knowledge and its lemma of random dynamical systems.(2) In the second chapter, Firstly, the background of partial dissipative stochastic reaction diffusion equations is introduced, and then the additive white noise of the original equations is eliminated by the transformation of stochastic processes. The only continuous solution of the equations is characterized as the corresponding continuous stochastic dynamical systems. Finally, it is proved that the existence and uniqueness of random attractors in space L~2 (R~n) ×L~2 (R~n) and the upper semi-continuity of the random attractors are discussed.(3) In the third chapter, Firstly, the background of the modified Swift-Hohenberg equation with non autonomous terms and random external force is introduced. And then the unique continuous solution of the equation is characterized as a continuous cocycle. Finally, it is proved that there exists a unique l(35)-pullback attractor in thespace of L~2 (R~n).