Mathematical Sciences: Topics in Partial Differential Equations and Their Applications

数学科学：偏微分方程专题及其应用

• 批准号: 9622867
• 负责人: Xuefeng Wang
• 金额: $ 3.93万
• 依托单位: Tulane University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622867&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topics; Partial; Differential

9625830 Wang The proposer plans to conduct his research in the following topics: (i) Convolution models for phase transitions; (ii) Concentration phenomena of solutions of semilinear elliptic equations; (iii) Qualitative behavior of solutions to chemotaxis systems. In the first front, effort will be made to understand the propagation of interfaces, the shape of energy minimizers and the dynamics of typical solutions. In the second front, of main concern are the existence and locations of concentration of solutions as a physical parameter becomes small. In the third front, the effects of various biological and physical parameters on the solutions of the systems will be studied. %%% The convolution models arise in the mathematical studies of grain boundaries in crystalline solids, binary alloys as well as in the study of excitation in neural networks. The equations in the second front are nonlinear Schr\"odinger equations of mathematical physics and one arising in the study of chemotaxis of cells. The chemotaxis systems considered in the third front model the oriented motion of cells in response to the concentration gradient of chemicals in the environment, and the research done and proposed here aims at understanding how the motility and the sensitivity of cells to the chemical affect the size of the population of cells as well as the way in which cells are distributed in the long run. The proposed research in all the three parts will help reveal and understand the important phenomena such as phase seperations, concentration and blow-up in the related fields of material sciences, biophysics and quantum mechanics. ***

拟在以下方面进行研究：(i)相变的卷积模型；半线性椭圆方程解的集中现象；（iii）趋化系统溶液的定性行为。在第一个方面，将努力理解界面的传播，能量最小化的形状和典型解决方案的动力学。在第二个方面，主要关注的是当物理参数变小时溶液浓度的存在和位置。在第三个方面，将研究各种生物和物理参数对系统解的影响。卷积模型出现在结晶固体、二元合金的晶界的数学研究中，也出现在神经网络的激励研究中。第二阵线的方程是数学物理中的非线性薛定谔方程和在细胞趋化性研究中出现的方程。趋化系统在第三前沿模型中考虑了细胞对环境中化学物质浓度梯度的定向运动，这里所做和提出的研究旨在了解细胞对化学物质的运动性和敏感性如何影响细胞群体的大小，以及细胞在长期内的分布方式。这三个部分的研究将有助于揭示和理解材料科学、生物物理学和量子力学等相关领域的重要现象，如相分离、浓度和爆炸。＊＊＊