Random Attractors For The Semi-linear Degenerate Parabolic Equations With Multiplicative White Noise In L2(D)

This paper mainly studies the existence of a random attractor in L2(D)for such RDS which generated by the semi-linear degenerate parabolic equations with mul-tiplicative white noise.The paper investigate the following semi-linear degenerate parabolic equations with multiplicative white noise equation where D∈RN(N≥2)is a bounded domain with smooth boundary δD, λ、c are positive constants,W(t)is a two-sided real-valued Wiener-process on a probability space(D,F,P). The diffusion coefficient σ(x),the nonlinearity f(·) satisfy the following assumptions.(Hα)The function σ:D�'R+∪{0}is measurable such that σ∈Lloc1(D)and for some α∈(0.2),lim infx�'z|x-z|-ασ(x)>0for every z∈D.(F)To the function f∈C1.(R,R),there is a number p≥2such that for all u∈R, f(0)=0,f’(u)≥-l f(u)u≥C1|u|p-K1|f(u)|≤C2|u|p-1+K2 where Ci,Ki(i=1,2) and l are positive constants.The paper contains four parts:In chapter one. introduces the background and the research status of RDS, random attractor and semi-linear degenerate parabolic equations, this paper’s main contents and significance,also provides some relevant basic knowledge.In chapter two, by using v(t)=e-cz(0,w)u(t) to simplify the equation, then by the use of Galerkin method proves that the semi-linear degenerate parabolic equations with multiplicative white noise has a unique solution, which generates a continuous RDS(θ,φ).In chapter three, formulate our main result that the RDS(θ,φ) has the only random attractor in L2(D).First, there is a absorbing set �'=B(ω)ω∈D for RDS(θ，φ) in L2(D) is got.Second, there is a absorbing set B(0. R(ω)) for R,DS(θ,φ) in D01(D) is proved.The last. according to embeddedness of Lemma1.3.1, we know that the a bounded set of D01(D) in L2(D) is compact,combined with Theorem1.3.1and the above proved can get that the RDS(θ,φ) has the only D-random attractor A(ω) in L2(D).In chapter four, some problems to be further studied are discussed.