拟线性抛物方程及微机电系统新动力学模型的基础理论研究

项目名称： 拟线性抛物方程及微机电系统新动力学模型的基础理论研究

项目编号： No.11471339

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 管理科学

项目作者： 邢瑞香

作者单位： 中山大学

项目金额： 45万元

中文摘要： 本项目研究一类拟线性抛物问题解的各种奇性和与稳态椭圆方程的关系。所关注的方程来自有几何、物理或工程背景的模型问题，如毛细理论，超导现象，微机电系统控制器等。由于这类问题的稳态椭圆方程特殊的结构，它的多解问题用标准的临界点理论难以解决，相应函数空间中的下半连续性和紧性如PS条件难以验证。另一种思路是通过相应抛物问题的解（抛物流）来研究椭圆多解问题，这有两种途径，一是研究抛物流的动力系统不变流形的结构，另一种是用抛物流代替伪梯度流来建立关键的形变引理，从而恢复临界点理论的使用。无论哪种办法，都需要对相应抛物流的奇性行为有好的理解。在此项目中，我们将以几个模型问题为主，调查相应抛物流的各种奇性行为，如解的爆破和熄灭现象，包括有限时间和无穷时间，解本身和梯度的爆破，解的爆破和熄灭集，精细爆破和熄灭速率，全局解的收敛性等基本理论问题。其中梯度爆破现象以及多种奇性同时发生是此类问题不同于半线性情形。

中文关键词： 临界点理论；形变引理；拟线性方程；热流方法；微机电系统

英文摘要： In this program, we are concerned with various singularities of solutions for some quasilinear parabolic equations and the relations with the corresponding stationary equations. These equations come from some important models having geometric, physical or engineering backgrounds such as capillary theory, superconductivity, Micro-Electromechanical System (MEMS), etc. The typical examples are of prescribed mean curvature equations of parabolic type. Since the classical critical points theory is hard to apply to multiple solution problems of the corresponding stationary problems, we here consider to use the methods of parabolic flows. To this end, we need to have a good understand about singularities of the flows. We mainly focus on several important model problems and study the singularities of the related parabolic flows such as blow-up and quenching phenomena, involving the solution and it's gradient, including infinite or infinite time, the set of singular points and the refined rates, the convergence of the global solution, etc. The singularities arising in these problems are more complicated than those in semilinear models.

英文关键词： Critical Point Theory;Deformation Lemma;Quasilinear Equation;Heat Flow Method;Micro-Electromechanical Systems(MEMS)