The Existence Of Random Attractors For Stochastic Plate Equations

Problem(1.1)models transversal vibrations of thin extensible elastic plate in a histo?ry space,which is established based on the framework of elastic vibration by Woinowsky-Krieger([1])and Berger([2]).It can also be regarded as an elastoplastic flow equation with some kind of memory effect.Subsequently,the long-term behavior of solutions for the deterministic plate equations have been extensively investigated.However,it is not avoid that the plate systems are disturbed by random factor in reality.Therefore,it is necessary and meaningful to study the stochastic plate equations.In this paper,we mainly focus on the asymptotic behavior of solutions for stochastic plate equations with additive noise.Details are as follows.Ⅰ.We first make a survey on the connections and differences between infinite dimen-sional dynamical systems and random dynamical system,simply precent the background and development of the plate equations.Ⅱ.Applying a prior estimate and operator decomposition method to obtain the existence and upper semicontinuity of random attractors for the extensible plate equations with critical exponent and additive noise(?)(?)(?)(?)where U is a bounded open set of R5 with a smooth boundary(?)U,α,β are posi-tive constant,p ∈R,u = u(x,t)is a real-valued function on U x[0,+∞),hj(x)∈(?)are independent two-sided real-valued Wiener processes on(Ω,(?),P).Ⅲ.Firstly,Making use of the methods of Shengfan Zhou[Nonlinear Anal.2015],we study the existence of random attractors for plate equations with memory and additive noise(?)(?)(?)(?)Secondly,by defining the energy functionals and using the compactness translation theorem,we prove the existence of random attractors for the continuous random dynami-cal systems generated by stochastic weakly dissipative plate equations with linear memory and additive noise(?)(?)(?)where U is a bounded open set of R5 with a smooth boundary(?)V u = u(x,t)is a real-valued function on(?)is a given external force.(?)are independent two-sided real-valued Wiener processes on(Ω,(?),P),μ(s)is the memory kernel.IV.In line with the methods of Meihua Yang[Nonlinear Anal.2011],we study the existence of random attractors for plate equations with displacement dependent damping and additive noise(?)(?)(?)(?)where U is a bounded open set of R5 with a smooth boundary(?)a real-valued function on(?)is a given external force.(?)are independent two-sided real-valued Wiener processes on(Ω,(?),P),σ(u)is the coefficient function of displacement dependent damping.