一类含有梯度项的奇异抛物及其椭圆问题的研究

项目名称： 一类含有梯度项的奇异抛物及其椭圆问题的研究

项目编号： No.11201311

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： 夏莉

作者单位： 广东财经大学

项目金额： 22万元

中文摘要： 本项目旨在对一类含有梯度项的奇异抛物及其椭圆问题进行基础理论研究。这类问题在物理、几何、化学及生物学等领域都具有很强的应用背景。由于方程中可能出现多处奇性和非线性，在研究的过程中需要根据方程本身的特点寻找不同的理论工具及研究方法，处理起来将有一定的难度和挑战。因此，本项目的研究意义在于发展和充实了奇异型偏微分方程理论，并且对上述领域中出现的实际问题提供理论上的支持和应用上的指导。 本项目研究内容主要包括以下三个方面：(1)含梯度项的奇异抛物问题的基本理论研究，包括弱解的合理定义，古典解与弱解的存在性，解的正则性、唯一性与多解性等；(2)含梯度项的奇异抛物问题解的行为，包括解的爆破、猝灭及解的渐近行为等；(3)某些含梯度项的奇异椭圆问题非负解的存在性等。

中文关键词： 梯度项；奇异性；抛物方程；存在性；渐近行为

英文摘要： This project is focused on the basic theory of a class of singular parabolic and elliptic problems with gradient term. These problems have strong application background in the domains of Physics, Geometry, Chemistry and Biology. Since there may be much singularity and nonlinearity in the equations, during the process of study we have to search different tools of theory and methods of research according to characteristics of equations, which will lead to certain difficulty and challenge. Hence, the significance of this project is to develop and enrich theory of singular partial differential equations, and give support from theory and guidance from application on practical problems emerging in the above domains. The contents of this project are maily composed of following three aspects: (1)basic theory of singular parabolic problem with gradient term, including proper definitions on weak solutions, existence of classical solutions and weak solutions, regularity, uniqueness and multi-solution property of the above solutions; (2)behaviour of solutions for singular parabolic problem with gradient term, including blow-up, quenching and asymptotic behavior of solutions; (3)existence of nonnegative solutions for some singular elliptic problems with gradient term.

英文关键词： gradient term；singularity；parabolic equation；existence；asymptotic behavior