Mathematical Sciences: Nonlinear Partial Differential Equations and Systems

倪9401333 该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。 该研究继续进行了几个项目，如数学生物学中的尖峰层研究。 这些都出现在数学描述的趋化聚集阶段的细胞黏菌。 一个是感兴趣的解决方案的相关方程表现出尖峰图案依赖于方程系数。 这种模式已经在实验中观察到。 第二条调查路线涉及种群动态中的交叉扩散. 这里要研究的主要问题是确定和理解这样的系统的稳定状态。 工作也将完成测量一类半线性热方程在n维空间中的稳定性和非齐次半线性椭圆或抛物方程中产生的无限粒子系统的概率。 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建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。

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.