| In this paper, we study the asymptotic behavior of solutions for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities with multi-plicative noise, additive noise defined on the periodic boundaries, and the stochastic semi-linear degenerate parabolic equations with additive noise in a bounded domain D (?) Rn. The random dynamical system generated by the corresponding problem is shown to have a random attractor in E0=H1x L2and Lq ((?) q≥2) respectively.This paper is divided into five:Chapter one: We recall the background on the theory of random dynamical systems, random attractors, and the research related to these equations, main con-tents of this paper, as well as some preliminary definitions and results, which will be used in this paper.Chapter two: We consider the dissipative Hamiltonian amplitude equation gov-erning modulated wave instabilities perturbed by additive noise: where u is an unknown complex valued function, i is the unit of imaginary number, the internal I=(-L, L),α,β and γ are positive constants, which satisfy β<γ, the functions hj∈H2(I), j=1,2,…, m, are time independent, the random functions Wj, j=1,2.…, m, are independent two-side real-valued Wiener processes on a probability space (Ω, F, P), and f(s) is C1, sf(s) is C2real valued function which satisfies that where0<∈<1,γ0is a constant depended on δ and∈, and F(s)=∫TS f(t)dt.By introducing two functions and a process, and the decomposition of solution semigroup, we have the main result of this chapter: the existence of a random at-tractor in E0.Theorem2.6.3. Assume hj∈D(A)=H1∩H2, then the random dynamical system φ(t,ω) modeling the dissipative Hamiltonian amplitude equation govern-ing modulated wave instabilities with additive noise possesses a compact random attractor A(ω), which attracts all bounded sets of E0=H1x L2.Chapter three: We consider the dissipative Hamiltonian amplitude equation governing modulated wave instabilities with multiplicative noise:The main result of this chapter is:Theorem3.5.3. The random dynamical system φ(t,ω) generated by the dissipa-tive Hamiltonian amplitude equation governing modulated wave instabilities with multiplicative noise possesses a compact random attractor A(ω), which attracts all bounded sets of E0=H1x L2.Chapter four: We consider the stochastic semi-linear degenerate parabolic e-quations with additive noise: where D (?) Rn(n≥2) is a bounded open set with smooth boundary (?)D,λ>0,{ωj}j=1m are independent two-side real-valued Wiener processes on a probability space (Ω,(?), P), and the diffusion coefficient σ(x), the nonlinear term g(·), and the functions{hj}j=1m satisfy the following hypotheses:(Hσ) The function σ:Dâ†'R is a non-negative measurable function such that σ∈Lloc1(D) and for some α∈(0,2), liminfxâ†'z|x-z[-ασ{x)>0for every z∈D;(F) The function f∈C1(R, R) satisfies a polynomial growth condition, that is, there exist C0, c1and large number c2, such that there exists M>0such that f(x)≥c2x2p-1if x≥M, while f(x)≤c2x2p-1if x≤-M.(H) The functions hj∈Lq(D)∩Dom(A)∩Dq(A), y=1,2,…, m, where Au=-div(σ{x)(?)u), Dom(A)={u∈D01(D,σ):Au∈L2(D)}, and Dq(A)={u∈D01(D,σ):∫D|Au|qdx<+∞}.Noting that a polynomial of odd degree with a positive leading coefficient sat-isfies the hypothesis (F).By the induction hypothesis of q, and using an asymptotic a priori estimate for far-field values of solutions, we get the final conclusion of this chapter:the existence of a random attractor in Lq ((?)q≥2).Theorem4.6.1. Assume that D (>) Rn is bounded and (Hσ)-(F)-(H) hold. Then the random dynamical system φ(t, ω) generated by the stochastic semi-linear degenerate parabolic equation with additive noise (4.1.1) possesses a random at-tractor Aq(ω) in Lq for any q≥2, which is a compact and invariant tempered random set attracting all tempered random subsets of L2under the Lq-norm topol-ogy. Furthermore, Aq(ω):=(A) for all q≥2, where A(ω) is the usual attractor in L2.Chapter five:Further studies are needed about these questions. |
PDF Full Text Request