完全非线性抛物方程定性理论及其应用

项目名称： 完全非线性抛物方程定性理论及其应用

项目编号： No.11201502

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 李静

作者单位： 中央民族大学

项目金额： 22万元

中文摘要： 本项目针对来源于渗流，燃烧，爆炸和电击穿等物理现象以及控制理论和金融领域的完全非线性抛物方程，旨在研究具有重要物理意义的定性理论。 本项目研究的完全非线性方程与线性模型的自由边界问题以及随机微分方程，二人博弈密切相关，拟解决的关键问题包括解的稳定性，奇异性和渐近性，其中很多都是人们所关注的热点问题。如自由边界问题多维行波解的稳定性，自由边界的等待时间，聚焦现象以及解的渐近复杂性问题。方程的完全非线性性，奇异性等均来源于实际问题，同时给我们的研究带来了本质性的困难。因此我们既需要经典的数学理论和研究工具，也需要发展新的研究思路和手段。我们的研究结果和方法将在一定程度上丰富偏微分方程理论并对解释某些物理现象提供重要参考。

中文关键词： 完全非线性；非局部；行波解；渐近性；极限集

英文摘要： This project focuses on fully nonlinear parabolic equations from physical phenomena in filtration, combustion, explosion and electrical breakdown, as well as control theory and financial area. It aims at studying qualitative theories with important physical significance. The fully nonlinear equations we study in this project are closely related to free boundary problems of linear models, as well as stochastic differential equations and two-person games. Key problems we plan to solve include the stability, singularity and asymptotic behavior of solutions, many of which are hot issues that people are concerned about, such as the stability of multidimensional traveling wave solutions, waiting time and focusing phenomenon of free boundary, as well as asymptotic complexity of solutions. The fully nonlinearity and singularity of equations come from practical problems, but brought difficulties to our research. So we need not only classical theories and research tools, but also to develop new ideas and methods. Our results and methods will enrich the theory of partial differential equations to a certain extent and provide important reference to explain certain physical phenomena.

英文关键词： fully nonlinear；nonlocal；travelling wave；asymptotic；limit sets