时间分数阶Fokker-Planck方程可以用于描述具有外力场的次扩散，它在高分子聚合物、Lévy飞行等领域有着广泛的应用。由于模型中的外力场是依赖时间和空间的，因而研究时间Fokker-Planck方程解的性质是一项挑战性工作。已有的工作大多数都是采用Galerkin逼近方法进行分析。本项目将研究时间分数阶Fokker-Planck方程初值问题适度解的适定性、初边值问题适度解和经典解的存在性以及倒向问题适度解的存在性和正则性。特别地，我们拟运用算子半群理论和Fourier分离变量方法来推导不含时间分数导数算子的适度解形式；探讨经典解的“工作空间”对参数α取值范围的影响，避免对α∈(1/2,1)的限制；以及揭示倒向问题在非线项不满足Lipschitz或者更高正则性的假设条件时适度解存在的机理。本项目旨在为分数阶偏微分方程的数值仿真和实际工程应用提供可靠的理论依据。

Time fractional Fokker-Planck equations are widely used in the fields of polymers and Lévy flight, which can be used to describe subdiffusion with general forcing. Since the forcing depends on the time as well as the space, it is a challenging task to study the properties of solutions of time fractional Fokker-Planck equations. In most existing literature, Galerkin approximation methods were used extensively. This project intends to study the well-posedness of mild solutions for the initial value problems for time fractional Fokker-Planck equations, the existence of mild and classical solutions for the initial boundary value problems, and then, the existence and regularity of mild solutions for the backward problems will be obtained. In particular, we will deduce the formula of mild solutions without time fractional derivative operators via the operator semigroup theory and Fourier method of separation of variables. The influence of the “working space” on the value range of the parameter α will be discussed and the restriction on α∈(1/2,1) can be avoided in the study of classical solutions. The existence of mild solutions of the backward problems will be discussed when the non-linearity term does not satisfy Lipschitz or higher regularity conditions. The completion of this project will provide a reliable theoretical support for numerical simulation and practical engineering application of fractional partial differential equations.