带多值算子的非线性抛物型方程的能控性

本项目拟对由多值算子所支配的非线性抛物型方程的能控性进行研究。带多值算子的非线性抛物型方程是偏微分方程中较为广泛的一类，许多数学物理问题可以转化为该类问题。本项目主要就退化的抛物型方程和带有极大单调图映射的抛物线方程的能控性进行探讨。其中，本项目将给出退化的抛物型方程的能控性条件；证明数学物理中经典的两类偏微分方程- - 障碍问题和Stefan问题- - 的能控性；研究带非线性边界条件的抛物型方程的能控性；也将从更高的角度提出Banach空间中抽象的非线性发展方程的能控性条件。本项目所研究的问题有广泛的工程应用背景，其理论的发展对于实践有重要的指导意义。在理论上，该类问题也具有一定的难度和挑战性，所得到的结果是偏微分方程的能控性的重要的补充.

The project concerns the controllability of nonlinear parabolic equations with multi-valued operators, which are so popular in the partial differential equations that they are related with many mathematical physics problems. In this project we focus on the controllability of parabolic equations with degenerate conditions and with multi-valued monotone graph. We will establish the controllability conditions for the degenerate parabolic equations; the controllability of the obstacle problems and Stefan problems; the controllability of parabolic equations with nonlinear boundary conditions; the controllability conditions of abstract nonlinear evolution equations in Banach spaces. The problems above are very important in applications for engineering. The results we get in this project will be useful completion of the controllability of partial differential equations. However, these problems cause difficulties and challenges.