Upper Semicontinuity Of Random Attractors For Stochastic Reaction-Diffusion Equation With Multiplicative Noise

In this paper, we study the asymptotic behavior of a stochastic reaction dif-fusion equation with multiplicative noise defined in bounded domains.Firstly, with a given nonzero random disturbance, the random dynamic systems generated by the unique solution of stochastic reaction diffusion equations exist a random attrac-tor in L2 space. Secondly, a family of random attractor under the L2 topology is upper semicontinuity, when the random disturbance approaches zero. The paper investigate the following stochastic reaction diffusion equation with multiplicative noise du+(λu-△u)dt=(f(x,u)+g(x))dt+ευοdW. O (?) Rn is a bounded set with a smooth bound in. The initial condition:u(x,0)= uo(x). where x ∈ (?), t≥ O, ε, λ are positive constants, g ∈ L2(?), W(t) is a two-sided real-valued Wiener process on a probability space and f is a nonlinear function satisfying the following conditions:For all x ∈ O, s ∈ R, f(x,s)s≤-α1|s|p+ψ1(x), |f(x,s)s|≤-α2|s|p-1+ψ2(x),where α1, α2 and β are positive constants, ψ1 ∈ L1(?) ∩ L∞(?), and ψ2 ∈ L2(?) ∩ Lq(?) withThis article is divided into four sections:In chapter 1, we briefly introduce the concepts of the random attractor and the random dynamic system. And we also show the background of stochastic re-action diffusion equation, some current researches in the world and the meaning of researching the upper semicontinuity of a random attractor.In chapter 2, we study a random dynamical system generated by the only solution of the stochastic reaction diffusion equation with multiplicative noise. By introducing a new variable and eliminating the random disturbance of the equation, we get the existence of a unique solution of stochastic reaction diffusion equation in bounded L2 space and then generate the random dynamic systems.In chapter 3, according to the theorem of existence of random attractor, we get the unique random attractor in bounded L2 space by proving the L2 space and the H10 space exist a random attraction set and Sobolev compact embedding H10 to L2.In chapter 4, according to the existence conditions of upper semicontinuity of random attractor, we prove the random dynamic systems tend to deterministic semigroup as the random disturbance tending to zero, when the initial values of solution of random equation approach those of deterministic equation. Secondly, we prove the union of the family of random attractors is compacted in L2 space, when the random disturbance fluctuates in a small interval; Finally, when the random disturbance approaches zero, we prove the existence of the upper bound of the family of the random attraction set.