非线性项不满足Lipschitz条件的微分方程在控制系统、生存理论、经济动力学和生态学中有很多应用，随着应用领域的不断拓展，多值动力系统的渐近行为成为重要前沿课题，而相应的非自治随机系统的研究工作还很鲜见。本项目将深入研究三类具有代表性的解不唯一的非自治随机无界时滞抛物型方程、非经典扩散方程、带有超临界Sobolev增长指数非线性项的波方程拉回吸引子的存在性、正则性和可测性，及非自治随机摄动和奇异摄动下更高正则性空间中吸引子的上半连续性。深入研究梯度类型的多值非自治和/或随机动力系统拉回吸引子的Morse分解理论，及参数摄动下Morse集的上半连续性和应用。近两年项目组成员给出了分数阶物质积分的数学定义、分析了其性质，它是描述反常运动泛函分布的数学工具，本项目还将继续研究均方收敛意义下带有分数阶物质导数的非自治随机一阶格点系统和无界时滞偏微分系统解的全局存在性与长时间行为，并进行数值模拟。

Differential equations with non-Lipschitz nonlinearities arise naturally from control systems, survival theory, economic dynamics and ecology. With the development of applications, the asymptotic behaviour of multi-valued dynamical systems has become a frontier research topic, however there is little reference on nonautonomous random dynamical systems, even in the single-valued case. The project will intensively study the existence, regularity and measurability of pullback attractors for nonautonomous and stochastic parabolic equations with unbounded delays, nonclassical diffusion equations and wave equations with supercritical exponent and without uniqueness, and the upper semicontinuity of pullback attractors for multi-valued nonautonomous random dynamical systems with nonautonomous and stochastic perturbation or singular perturbation in the space with higher regularity. Morse decomposition theory of pullback attractors for gradient-like multi-valued nonautonomous and/or random dynamical systems, the upper semicontinuity of Morse sets and applications will be studied. In recent two years, the members of the project team gave the mathematical definition of the fractional substantial integral and analyzed its properties. Since the fractional substantial derivative is an important mathematical tool to describe functional distribution of the anomalous motion, this project will continuously study the global existence and long time behaviour of solutions for nonautonomous and stochastic first-order lattice systems and unbounded delay partial differential systems with fractional substantial derivative in the sense of mean-square convergence, and the numerical simulation also will be concerned.