本项目旨在运用KAM理论研究时滞反应扩散方程和时滞波动方程拟周期解的存在性及其相关性质，主要困难在于克服时滞效应对于方程动力学的影响，以及拟周期解构造中出现的小分母问题。一方面，拟运用相空间分解技巧并结合经典的KAM方法研究时滞反应扩散方程不变环面的存在性和可积标准形，将KAM方法研究范围扩展到具有空间扩散的偏泛函微分方程上去，特别是进一步研究切向频率与法向频率的共振问题而使拟周期解理论更加完善。另一方面，拟采用KAM理论中的Craig-Wayne-Bourgain方法来构造时滞波动方程的拟周期解，为以后研究高维哈密顿偏微分方程的拟周期解等相关问题打下基础。通过本项目研究，既丰富和完善偏泛函微分方程定性理论和动力学性质，又扩展KAM理论的应用范围，且能使不同的数学问题相互融合，还可以为应用领域的工作者提供一些可靠的理论依据和解决问题的方法。

The goal of this project is to study the existence, as well as some associated properties, of quasi-periodic solutions for delayed reaction diffusion equations and delayed wave equations by the KAM theory. The main difficulties are to overcome the effect of time delay on the dynamics and the small divisor problem in the construction of quasi-periodic solutions. One the one hand, for delayed reaction diffusion equations, we combine the technique of the phase space splitting with the traditional KAM method to establish the existence of the invariant torus and its integral normal form, which will generalize the application of the KAM method to partial functional differential equations with space diffusion. In particular, we shall study further the resonance between the tangent frequency and the normal frequency to improve the existing quasi-periodic theory for delayed reaction diffusion equations. On the other hand, for delayed wave equations, we employ the Craig-Wayne-Bourgain method in the KAM theory to construct directly the quasi-periodic solutions, from which we will have a deep knowledge of the method to engage in the prospective study of higher dimensional Hamiltonian PDEs. Our results not only enrich the qualitative theory and the dynamics of partial functional differential equations, but also extend the application fields of the KAM theory. Most importantly, the study involves the interaction of different branches of mathematics, and provides a solid theoretical foundation and some useful methods for researchers in applied fields.