非自治随机 Boissonade 系统在不同边界条件下的渐进行为

The problems concerning the attractor of infinite-dimensional random dynamical systems are now important and frontier topics in the field of international dynamical systems. Boissonade is one of the important mathematical models in describing Turing pattern. This project will mainly study the following several problems concerning the random attractor of nonautonomous random dynamical system and asymptotic behaviors of the nonautonomous stochastic Boissonade system: the existence, upper semi-continuity, upper bound of fractal dimension of random attractors and the existence of random exponential attractors for Boissonade system effected by time-dependent deterministic forces and white noise with different boundary conditions and on unbounded domains; method to estimate the lower bound of fractal dimension and conditions for the lower semi-continuity of random attractors of nonautonomous random dynamical systems and their applications in nonautonomous stochastic Boissonade system; relationship between the parameters of nonautonomous stochastic Boissonade system and upper and lower bounds of fractal dimensions of its random attractor; relationship between the parameters of nonautonomous stochastic Boissonade system and upper and lower semi-continuity of its random attractors. The realization of this project will enrich the theory of random dynamical systems and its applications, and provide reference to study of asymptotic behaviors of other pattern reaction-diffusion systems.

无穷维随机动力系统的吸引子相关问题是当前国际动力系统研究的热点和前沿课题。Boissonade系统是刻画图灵斑图的重要的数学模型之一。本项目主要研究非自治随机动力系统的随机吸引子和非自治随机Boissonade系统的渐进行为的若干问题：研究不同边界条件下以及无界域上Boissonade系统同时受到依时间确定性外力和白噪声扰动时的随机吸引子的存在性、上半连续性、分形维数上界以及随机指数吸引子的存在性；研究非自治随机动力系统的随机吸引子的分形维数下界的估计方法和下半连续性的条件及其在非自治随机Boissonade系统上的应用；研究非自治随机Boissonade系统的参数与其随机吸引子的分形维数上、下界的关系；研究非自治随机Boissonade系统的参数与其随机吸引子的上、下半连续性的关系。本项目内容的实现将丰富随机动力系统的理论及其应用，给其他斑图反应扩散系统的渐进行为研究提供参考。

本项目主要研究了非自治随机Schrodinger格点系统、非自治随机Boissonade系统在依时间确定性外力和白噪声扰动下的指数吸引子的存在性、分形维数上界估计等问题。我们得到了具时变耦合系数的Schrodinger格点系统受依时间外力影响下的拉回指数吸引子、拟周期外力驱动下的非自治Schrodinger格点系统的一致指数吸引子的存在性和分形维数上界，此上界与系统的各个参数关系。Schrodinger谔格点系统相当于空间变量取值为无界的整数集，对非自治随机Schrodinger格点系统和无界域上非自治随机Boissonade系统的指数吸引子研究提供一定的参考价值，也为其它格点系统的指数吸引子研究提供依据，并为后续研究积累了丰富的经验。