Mathematical Sciences: Nonlinear Partial Differential Equations and Systems

本奖项支持的工作重点是非线性偏微分方程、系统及其应用的数学分析。这项研究还在继续进行几个项目，比如对数学生物学中的尖峰层的研究。这些出现在细胞黏菌的趋化聚集阶段的数学描述中。一个感兴趣的是相关方程的解，它表现出依赖于方程系数的尖形模式。这种模式已经在实验中观察到。第二项调查涉及人口动态中的交叉扩散。这里要研究的主要问题是确定和理解这类系统的稳定状态。我们还将研究在n维空间中测量一类半线性热方程的稳态稳定性，以及在概率无限粒子系统中产生的非齐次半线性椭圆型或抛物型方程。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。

Ni 9401333 Work supported by this award focuses on the mathematical analysis of nonlinear partial differential equations, systems and their applications. The research continues several projects such as the study of spike-layers in mathematical biology. These arise in mathematical descriptions of the chemotactic aggregation stage of cellular slime molds. One is interested in solutions of the relevant equations which exhibit spiky patterns dependent on the equation coefficients. Such patterns have been observed experimentally. A second line of investigation concerns cross- diffusion in population dynamics. The primary question to be studied here is one of determining and understanding the steady states of such systems. Work will also be done measuring the stability of steady states of a class of semilinear heat equations in n-dimensional space and on the inhomogeneous semilinear elliptic or parabolic equations arising in infinite particle systems in probability. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.