【作者】 王刚；

【导师】 汤燕斌；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2014， 博士

【摘要】 确定性的反应扩散方程在斑图理论,种群动态演化等研究中取得了很大成功.例如,对细胞和神经等复杂系统和网络的研究,导致了数学生物学的诞生.但自然界中的各种系统都有可能受随机外力,随机介质,随机边界条件和随机环境等因素的干扰和影响.近一二十年来,随机反应扩散方程越来越受到研究人员的关注.本文主要研究几类随机反应扩散方程的渐近行为(最后一章也研究了确定性方程的渐近性质),内容包括：无界域上带加性噪声随机反应扩散方程的(L2,H1)-随机吸引子；有界域上带乘性噪声随机反应扩散方程的(L2,H10随机吸引子；随机不变集的分形维数；R3中有界域上一类确定性反应扩散方程在空间L2p-2(D)和H2(D)中的指数吸引子；一类带参数的非经典扩散方程指数吸引子的鲁棒性等.全文分为三个部分：第一部分主要介绍了无穷维动力系统和随机动力系统的发展进程.回顾了随机吸引子相关概念及已有方法和结论,同时也简要介绍了本文的研究内容.然后给出了本文中要用到的一些基本概念和结论.第二部分,是本文的核心内容.首先,研究了定义在全空间Rn上带加性白噪声的随机反应扩散方程的(L2,H1)-随机吸引子的存在性,其中非线性反应项满足任意的p-1(p≥2)次多项式增长条件.用截断方法和尾估计方法克服无界域上Sobolev嵌入的非紧性的困难,证明了(L2,H1)-渐近紧性.在利用截断法证明解在球内部的渐近紧性时,我们建立了一种新的估计,这种估计足够精细使得我们不需要对方程两边求导就可以得到解的渐近紧性.其次,证明了有界域D∈Rn上带乘性白噪声的随机反应扩散方程的(L2,H10)-随机吸引子的存在性.与有界域上加性噪声的情况相比,乘性噪声的情形更为复杂,需要对随机系数和Gronwall型不等式作更精细的估计.然后,给出了一个新的关于随机不变集分形维数的上界估计.这是确定性结论在随机情况下的推广.与确定性情形不同的是,随机不变集及其覆盖是随时间变化而变化的,所以这种推广并非是平凡的,我们利用Poincare回归定理来克服这一困难.所得抽象结论的优势在于它不需要系统的可微性,而且对Banach空间中的问题也成立,从而它有更广泛的应用.作为应用我们给出了一类随机半线性退化抛物方程的随机吸引子的分形维数估计.最后,研究了确定性反应扩散方程的指数吸引子.主要结果包括：(1)我们研究了一类R3有界域上带非齐次项g的反应扩散方程,其中的非线性项f满足任意的多项式增长条件.首先我们证明了其解半群在H2D)范数拓扑下存在指数吸引子.其次,给出了一个新的逼近结论并利用这个结论证明了对任意的g∈L2(D)解半群在L2p-2(D)范数拓扑下存在指数吸引子.(2)我们改进了历史文献中关于鲁棒指数吸引子族的存在性的抽象结论,将其中的光滑性进一步弱化.改进后的结论有更广泛的应用对象.利用这一结论,我们证明了一类带参数的非经典扩散方程在H10在鲁棒的指数吸引子族,其中初值也属于H10.第三部分,总结全文并提出一些今后有待进一步研究的问题.更多还原

【Abstract】 The deterministic reaction-diffusion equations have been successfully applied in pat-tern theory and the evolution of population dynamics. For example, the study of complex systems and networks such as cells and nerves led to the birth of mathematical biology. However, all kinds of dynamic systems can be influenced by natural or man-made random factors, such as random external force, random medium, random boundary conditions, ran-dom environments, etc. In recent decades, stochastic reaction-diffusion equations attract more and more attentions of researchers. In this doctoral dissertation, we study the asymp-totical behavior for several classes of stochastic reaction-diffusion equations(we also consid-er the asymptotical behavior for deterministic equations in the last chapter). These asymp-totical behavior includes:The (L2,H1)-random attractors for stochastic reaction-diffusion equations with additive noise on unbounded domains; The (L2,H1/0)-random attractors for stochastic reaction-diffusion equations with multiplicative noise on bounded domains; The fractal dimension of random invariant sets; The exponential attractors in L2p-2(D) and H2(D) for a class of deterministic reaction-diffusion equations, where D is a bounded set in R3; The robustness of exponential attractors for a class of nonclassical diffusion equa-tions with parameter.This thesis consists of three parts:In the first part, we introduce the evolution of infinite-dimensional dynamical systems and random dynamical systems. We recall the basic notations and the known methods and results related to random attractors, and briefly describe our research results of the present paper. Then, we present some preliminary definitions and results that will be used in this thesis.The second part is the core content of this thesis. Firstly, we study the existence of (L2,H1)-random attractors for a class of stochastic reaction-diffusion equations with addi-tive noise defined on the whole space Rn, where the nonlinearity is supposed to satisfy the polynomial growth of arbitrary order p-1(p>2). To prove the (L2, H1)-asymptotically compactness, we use the cut-off technique and the tail estimate method to overcome the difficulty of the lack of compactness of Sobolev embeddings on unbounded domains. We establish a new estimate when we use the cut-off technique, and the estimate is accurate enough so that we can obtain the asymptotically compactness in any bounded ball without differentiating the equation.Secondly, we prove the existence of (L2, H1/0)-random attractors for a class of stochastic reaction-diffusion equations with multiplicative noise defined on a bounded domain D C Rn. Compare with the case for additive noise on bounded domains, it is more complicated for multiplicative noise, and the estimates on random coefficients and the Gronwall type inequality should be more accurate.Thirdly, we give a new upper bound estimate on the finite fractal dimension of random invariant sets. This is the extension of the results of the deterministic case to stochastic case. The extension is non-trivial, since, different from deterministic case, the random invariant sets and their coverings are varied with time, and we use Poincare recurrence theorem to overcome this difficulty. This new abstract result do not require differentiable property of the system and can be applied to Banach spaces, hence it has widely applications. Then this result is applied to a stochastic semilinear degenerate parabolic equation and an upper bound is obtained for the fractal dimension of the random attractors.Finally, we do some research on exponential attractors for deterministic reaction-diffusion equations. These include:(1) We consider a class of reaction-diffusion equations defined on a bounded domain in R3with arbitrary polynomial growth nonlinearity/and nonhomogeneous term g. We first construct exponential attractors in H2(D) for the under-lying semigroup. Then, we obtain the exponential attractors in L2p-2(D) for any g e L2(D) by using a new approaching technique.(2) We improve the abstract results on the existence of robust exponential attractors in previous papers, and use a weaker smoothing condition in our abstract result. The modified result has more widely applications. As an application, we prove the robustness of exponential attractors in H1/0for a class of nonclassical diffusion equations with parameter, and the initial datum belongs to H1/0.In the third part, we summarize the main results and propose some problems for future research.更多还原