Mathematical Sciences: Studies on Nonlinear Partial Differential Equations

小行星9418779 该奖项支持在非线性偏微分方程的两个领域中出现的问题的数学研究。 第一部分是关于n维空间中半线性方程正解的渐近性态和分类的分析，主要讨论Lane-Emden方程。这个问题已经在各种特殊情况下进行了详尽的研究。 这项工作的扩展和推广也将进行。 正解的唯一性，特别是在径向解的情况下，也将采取。 一个新的研究方向关注半线性系统如Fujita系统的存在性和不存在性问题。 更多的应用研究将考虑热敏电阻问题，分析电阻随温度升高而迅速增加的电阻。 最后，我们将研究拟线性椭圆型方程基态的存在性问题和Monge-Amper方程的Dirichlet问题 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。 ***

9418779 Zou This award supports mathematical research on problems arising in two areas of nonlinear partial differential equations. The first is related to the analysis of the asymptotic behavior and classification of positive solutions of semilinear equations in n-dimensional space, focusing on the Lane-Emden equation. The problem has been studied exhaustively in various special cases. Extensions and generalizations of this work will also be carried out. Uniqueness of positive solutions, especially in the case of radial solutions will also be taken up. A new direction of research concerns existence and non-existence questions for semilinear systems such as the Fujita system. More applied studies will consider the thermistor problem, analyzing electrical resistors whose resistance increases rapidly with increasing temperature. Finally, work will be done on the existence problem for ground states for quasilinear elliptic quations and the Dirichlet problem for the Monge-Amper Equation 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. ***