本项目主要研究物理、力学等学科中出现的重要的非线性发展方程无穷维动力系统中以下三个核心问题：（A）研究无穷维动力系统的正不变吸引集的维数估计、指数耗散性理论及其在抛物型和耗散双曲型方程中的应用；（B）拓展全局（拉回）吸引子下半连续性理论及其在非自治半线性抛物型方程中的应用；（C）拓展拉回吸引子的分形维数下界估计理论及其在非自治半线性抛物型方程中的应用。对于（A），构造弱指数吸引子并用平衡点及指标理论研究维数估计；对于（B），构造非自治扰动系统的李雅普诺夫泛函来研究连续性；对于（C），研究分形维数和Z2指标的关系，将对拉回吸引子的Z2指标下界的估计转化为对其分形维数下界的估计。这些问题具有深厚的实际物理背景，也是近年来国际上无穷维动力系统理论上极为重要的、前沿的核心问题。这些问题深刻的研究成果，必将给无穷维动力系统的理论及应用的研究带来深刻的影响和很大的挑战性，具有重要的理论和应用意义。

This project will mainly study the following three core problems in the infinite-dimensional dynamical systems of important nonlinear evolutionary equations arising from subjects of physics, mechanics, and so on: (A) researches on the dimensional estimates of positive invariant attracting sets and exponential dissipative theories in infinite dimensional dynamical systems and their applications in parabolic equations and dissipative hyperbolic equations; (B) extensions of theories of lower semicontinuity of global (pullback ) attractors and their applications in non-autonomous semilinear parabolic equations; (C) extensions of the theories on lower estimates of fractal dimensions of pullback attractors and their applications in non-autonomous semilinear parabolic equations. For (A), construct the weak exponential attractors and use the equilibrium point and index theories to study dimensional estimates; for (B), construct the Lyapunov functional of non-autonomous perturbation system to study continuity; for (C), study the relationship between fractal dimension and Z2 index, and further transfer the lower estimate of Z2 index of pullback attractors into the lower estimate of its fractal dimension..These problems possess profound practical physical background and are also very active, important, frontier’s core problems in the theory of infinite dimensional dynamical systems in recent years. The deepgoing research achievements on these problems will necessarily bring about the deep impact on and great challenges of the researches of theories of infinite-dimensional dynamical systems, and possess important theoretical and practical significance.