Discussion On The Existence Of Random Attractors Of Some Stochastic Evolution Equations

Early in the 1980s, the conception of attractors has been introduced. It can describe the long time behavior of the dynamical system generated by the nonlinear dissipative evolution equations. Because the study of attractors involves nonlinear evolution equations which reflect many natural phenomenon and have strong practical background, so far the research about attractors is still quite active. The cases for many deterministic systems have been studied by many authors. However, in reality, it is inevitable that most systems are disturbed by random noise. The deterministic system model is only the ideal model of the actual system. stochastic systems can describe the nature in a more natural and true way. In this paper, we mainly discuss the existence of random attractors. we use the theory for the existence of random attractors of the quasi-continuous dynamical system which is established by Li [22] to prove the existence of random attractors for the reaction-diffusion equation with multiplicative noise and the P-Laplacian equation with multiplicative noise on Lp(D)(p≥2) respectively. And, furthermore, we use the tail-estimates method which is established by Wang[25] to prove the existence of the random attractors for the P-Laplacian equation with multiplicative noise on unbounded domains.In chapter 2, we discuss the reaction-diffusion equation with multiplicative noise: where D (?) Rn is a bounded open set with regular boundary (?)D, the constant number b>0 and the function f is a polynomial of odd degree with a positive leading coefficient:The white noise described by a process W(t) results from the fact that small irregularity has to be taken account in some circumstances. Here, we assume that W(t) is a two-sided Wiener process on the probability space (Ω, F, P), whereΩ={ω∈C(R, R):ω;(0)= 0}, F is the Borel sigma-algebra induced by the compact-open topology ofΩand P is a Wiener measure. For the stochastic dynamical system generated by the equation (1), we get the following result:Theorem 2.4: Suppose D (?) Rn and D is bounded. Then the stochastic dynamical system generated by the reaction-diffusion equation(1) has a random attractor in Lp(D)(p≥2).In chapter 3, we discuss the P-Laplacian equation with multiplicative noise: where D (?) Rn is a bounded open set with regular boundary (?)D, the constant number b> 0. the function f is a polynomial of odd degree with a positive leading coefficient. The white noise described by a process W(t) results from the fact that small irregularity has to be taken account in some circumstances. Here, we assume that W(t) is a two-sided Wiener process on the probability space (Ω,F.P), whereΩ={ω∈C(R,R):ω(0)= 0}, F is the Borel sigma-algebra induced by the compact-open topology ofΩand P is a Wiener measure.Firstly, we prove the continuous RDS which is generated by the P-Laplacian equation (2) has a compact random absorbing set, thus we get: Theorem 3.4:Suppose D C Rn is bounded, Then the RDSφgenerated by the stochastic P-Laplacian equation (2) possess a compact random attractor A2 in L2(D). That is A2 is compact, invariant in L2(D) and attracts all deterministic bounded sets of L2(D) in the topology of L2(D).Then, based on the theory for the existence of random attractors of the quasi-continuous RDS (see [22]), we prove that the RDS generated by the P-Laplacian equation (2) has a bounded random absorbing set and the dynamical systemφisω-limit compact, thus we get:Theorem 3.6:Assume D C Rn is bounded. Then the RDSφgenerated by the stochastic P-Laplician equation (2) possesses a random attractor Aq(ω) in Lq(D) for any q> 2.In chapter 4, we discuss Stochastic P-Laplacian equation with multiplicative noise on unbounded domains: du+(-div(|▽u|p-2▽u)+λu)dt= (f(x,u)+g(x))dt+cuοdW(t), x∈Q, t> 0 (3) with the boundary condition u|(?)Q= 0, and the initial condition u(x,0)= u0{x), x∈Q. where Q= D×Rn-1, D is a bounded domain in R,λ> 0, c> 0,p> 2 are constants, g∈L2(Q), and f is a nonlinear function satisfying the following conditions: f∈C1(Q×R,R)i; (ⅰ) f(x, u)u<≤-α1|u|p+ψ(x) for all x E Q and u∈R; (ⅱ) |f(x,u)|0, v>0 are given. The space-periodicity boundary condition areWe also consider the following condition as considered in [1]:Although this problem has been discussed by [1], they obtained, in fact, a attrac-tor in the subspace Ho of odd functions. We will prove the existence of a global attractor which attracts all bounded sets of the space L2(Ω), under the following assumption where L is the length ofΩand v is the given number defined in (4):Theorem 5.3:Assume (Ⅰ) is satisfied, then the semigroup{S(t)}t≥0 associated with Equation (4) possesses a global attractor A, which attracts all bounded sets of H.