非均匀介质中非线性拋物型方程的奇性分析

The nonlinear parabolic problems in non-uniform media, due to the spatial dependence of environmental factors involved, can be used to more accurately describe the rich nonlinear phenomena, and the related analysis and study must be challenging and interesting. This project will deal with the nonlinear parabolic equations with space dependence sources. The topics include the relationship between the asymptotic profile and the weight functions, effects of different types of source to the generation and propagation of singularities in solutions. In addition, the project will also study the parabolic problems with both weighted sources and weighted boundary conditions, and give the impact of the related weight functions to the blow-up behavior of solutions by establishing the maximum principle for the problems with non-uniform sources and boundary conditions. Our experiences of the studies for various nonlinear parabolic problems would benefit overcoming the essential difficulties resulted from the related weight functions. The results of the project will be helpful for better understanding the rich nonlinear phenomena in the nature.

非均匀介质中的非线性抛物问题考虑了空间依赖的环境因素，可以更精确描述现实中的各种非线性现象，相应的分析研究是一个充满挑战和趣味的探索。本项目拟研究源项为非各向同性的非线性拋物型方程和方程组，探讨解的 blow-up profile 与源的权函数因子的定量关系，揭示相关权函数对解的奇性产生与传播的影响；此外，本项目还将研究源项与边值条件均含有权函数的非均匀抛物问题，通过间接构建可容纳两类权函数因子的最大值原理，给出其对解的 blow-up 行为的影响。结合以往研究各类非线性抛物问题的经验，攻克由相关权函数带来的本质困难。研究成果将有助于深入理解和认识自然界中丰富的非线性现象。