带乘法扰动的反应扩散方程随机吸引子在p次可积空间的正则性与上半连续性
发布时间：2019-06-15 15:49
【摘要】：本文主要研究了带乘法扰动的反应扩散方程及其随机动力系统,随机吸引子的性质.通过对方程唯一解生成的随机动力系统及其(L2,Lp)-随机吸引子的一致渐近估计,我们证明了当扰动量处于正的有限区间时,随机动力系统在整个非负扰动区间的任何点都是上半连续的.我们考擦以下方程：其中x∈Rn,t≥0,u=u(x,t),初值条件为u(x,0)=u0(x).ε≥0,常数入是正的,g∈L2(Rn)∩LP(Rn),W(t)是概率空间(Ω,F,P)上的一个双边实值Wiener过程.对所有x∈Rn,u∈R,非线性函数f满足以下条件：其中：α1,α2和β是正的常量,本文一共分为四个章节：第一章,主要简述了随机吸引子和随机动力系统概念的产生及其对随机偏微分方程研究重要意义,然后介绍当下国内外对随机偏微分方程的研究现状,着重突出本文所做研究的意义,并简要阐述本文的研究内容与方法.第二章.引入与本文相关的关于随机动力系统和随机吸引子的基本定义与本文所需且已被证明的一些抽象结果和定理.第三章,本章节通过替代把随机反应扩散方程化成一个确定性偏微分方程,在利用其解的存在唯一性定理生成一个随机动力系统和(L2,Lp)-随机吸引子.最后得出该系统的上半连续性定理.第四章,根据随机动力系统的上半连续性的判定条件,首先证明随机动力系统在L2,p上的吸收性(引理4.1,引理4.2).为了证明Lp上系统在任意有限区间的渐进紧性(引理4.6),必须先证明三个辅助引理(引理4.3,引理4.4,引理4.5),最后我们证明随机动力系统在L2上的收敛性.从而定理得证.
[Abstract]:In this paper, we mainly study the properties of reaction-diffusion equation with multiplication disturbance and its stochastic dynamic system and random Attractor. Through the uniform asymptotic estimation of the stochastic dynamic system generated by the unique solution of the equation and its (L _ 2, L _ p)-random attractor, we prove that when the disturbance is in a positive finite interval, the stochastic dynamic system is upper semicontinuous at any point in the whole nonnegative disturbance interval. We examine the following equation: where x 鈮，
本文编号：2500332
[Abstract]:In this paper, we mainly study the properties of reaction-diffusion equation with multiplication disturbance and its stochastic dynamic system and random Attractor. Through the uniform asymptotic estimation of the stochastic dynamic system generated by the unique solution of the equation and its (L _ 2, L _ p)-random attractor, we prove that when the disturbance is in a positive finite interval, the stochastic dynamic system is upper semicontinuous at any point in the whole nonnegative disturbance interval. We examine the following equation: where x 鈮，
本文编号：2500332
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