Mathematical Sciences: Semilinear Elliptic Equations and Systems

数学科学：半线性椭圆方程和系统

• 批准号: 8801587
• 负责人: Wei-Ming Ni
• 金额: $ 10.23万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-06-15 至 1991-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8801587&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Semilinear; Elliptic; Equations

Work on this project combines geometrically motivated problems set in an analytic framework. It represents a continuation of research in four areas. In the first, work will continue in establishing existence and describing properties of conformal metrics with prescribed scalar or Gaussian curvature on complete manifolds. This is equivalent to finding positive solutions of an elliptic, nonlinear equation on the manifold, whose linear term is the Laplace-Beltrami operator in the metric with scalar curvature appearing as a multiplier of the unknown function. Although much is known about the compact case, relatively little has been done in the non-compact setting until recently. A second line of investigation will focus on developing a complete understanding of finite mass solutions and infinite mass solutions of the Matukuma equations in astrophysics. These are semilinear equations, first proposed in 1930, to model the dynamics of global star clusters. These equations are believed to be an improvement of Eddington's 1915 model. The unknown function here is the gravitational potential which, for the finite total mass case, has only recently been shown to have positive entire solutions (the corresponding Eddington equation has none). Many questions remain unanswered regarding the two models and discrepancies observed between their solutions. Work will also be done in determining whether the Lane- Emden equation of astrophysics has oscillatory as well as non- oscillatory solutions and analyzing the asymptotic behavior of singular radial solutions. In addition, relatively recent work on chemotaxis has led to new results on solutions of parabolic systems describing the oriented movement of cells in response to chemicals in the environment. Further work remains in determining how the cells move toward places of higher concentration and then aggregate.

这个项目的工作结合了几何动机 在一个分析框架中提出问题。 它代表了一 继续在四个方面进行研究。 首先，工作将 继续建立存在和描述属性 上具有指定标量或高斯曲率的共形度量 完备流形 这相当于发现阳性 流形上的椭圆型非线性方程的解， 其线性项是度量中的Laplace-Beltrami算子 标量曲率作为未知数的乘数出现 功能 虽然对紧凑型情况了解很多， 在非紧凑环境中所做的相对较少， 最近 第二条调查线将集中在开发一个 完全理解有限质量解和无限质量解 天体物理学中Matukuma方程的质量解。 这些 是半线性方程，首次提出于1930年，以模拟 全球星星团的动力学。 这些方程被认为 是爱丁顿1915年模型的改进 未知 这里的函数是引力势， 有限总质量的情况下，最近才被证明有 正整体解（对应的爱丁顿方程 没有）。 关于这两个问题仍有许多问题没有答案。 模型和他们的解决方案之间观察到的差异。 工作也将在确定是否车道- 天体物理学中的埃姆登方程既有振荡的，也有非振荡的 振动解的渐近性态 奇异径向解 此外，最近的工作 关于趋化性的研究导致了抛物方程解的新结果 描述细胞响应于 环境中的化学物质。 进一步的工作仍在 确定细胞如何向更高的位置移动 浓缩，然后聚集。