双曲流形上非线性扩散方程的若干定性问题

项目名称： 双曲流形上非线性扩散方程的若干定性问题

项目编号： No.11526052

项目类型： 专项基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 王智勇

作者单位： 福建师范大学

项目金额： 3万元

中文摘要： 本项目拟研究双曲流形上非线性扩散方程的若干动力学性质. 随着欧氏空间上偏 微分方程理论的不断完善, 2000年以来, 流形上的各类偏微分方程得到了国际学 术界越来越多的关注. 本项目重点研究双曲流形上热方程的指标理论, 生命跨度, 周期解. 双曲流形是负常截面曲率流形的代表. 它和欧氏空间, 球面一起构成了 等价意义下常截面曲率流形的完全分类. 研究双曲空间上的偏微分方程对于研 究一般的负截面曲率流形上的偏微分方程有重要的借鉴意义. 本项目试图发现双 曲流形上相关问题和欧氏空间上经典结果的联系和区别, 研究空间的几何性质对 热方程动力学性质的影响, 完善偏微分方程理论.

中文关键词： 非线性扩散方程；双曲空间；爆破；生命跨度；周期解

英文摘要： We study some qualitative problems of nonlinear reaction-diffusion equations on hyperbolic space. With the improvement of the theory of partial differential equations in the Euclidean space, since 2000, people have paid attention to partial differential equations on manifolds. This project focuses on Fujita exponents on nonlinear heat equation on hyperbolic manifold, life span problems, the existence or non-existence of periodic solutions. Hyperbolic space is a manifold with constant sectional curvature, which together with Euclidean space and spherical form Equivalent manifolds with constant sectional curvature in the sense of a complete classification. Studying partial differential equations in hyperbolic space for research general partial differential equations on manifolds with negative sectional curvature has important significance. This project tries to find the relationship and the distinction of the classical problems between the hyperbolic space and Euclidean space. Our results will show how the geometric of space to influence of heat equations, and our methods will enrich the theory of partial differential equations.

英文关键词： nonlinear diffusion equations；hyperbolic space；blow-up；life span；priodic solutions