Mathematical Theories of the Thermistor Problem

首席研究员将研究描述某些半导体中热传导和导电结合过程的数学模型。热导率和电导率被认为是高度依赖于温度的。由于描述这些过程的偏微分方程存在强退化性和二次非线性，这些物理问题在非线性分析中引起了深刻而有趣的问题。部分数学分析涉及到部分正则性的应用和奇点附近解的局部描述。然而，这些过程中的解具有丰富的结构，不同的结构需要不同的数学方法。如何控制微处理器的温度是半导体工业中一个非常重要的问题。计划中的研究将使用数学模型来揭示半导体中的温度如何依赖于已知数据。期望对这些问题的了解可以帮助工程师和研究人员设计未来的产品。

The principal investigator will study mathematical models that describe the combined processes of heat conduction and electrical conduction in certain semiconductors. Thermal conductivity and electrical conductivity are assumed to be highly temperature-dependent. Due to the presence of strong degeneracy and quadratic nonlinearities in the partiual differential equations that describe these processes, these physical problems lead to deep and intriguing questions in nonlinear analysis. Part of the mathematical analysis involves the applications of partial regularity and the local description of a solution near a singularity. However, solutions in these processes exhibit rich structures and different structures require different mathematical approaches.How to control temperature in microprocessors is a very important issue in the semiconductor industry. The planned research will use mathematical models to reveal how the temperature in a semiconductor depends on the known data. It is expected that the understanding of these issues can assist engineers and researchers in the design of future products.