Analysis of interface motion and blow-up phenomena in nonlinear partial differential equations

本课题主要研究方向为：(1)非线性扩散方程中的行波，(2)非线性扩散方程和界面运动的奇异极限，(3)非线性热方程的爆破，(4)三维细胞电生理模型的数学分析。主题(4)是在节目开始后添加的。在主题(1)中，我们研究了行波与障碍物的碰撞、平面波的稳定性以及行波速度最大化问题。在主题(3)中，我们研究了Sobolev超线性热方程的爆破分类，以及辫群理论在死核问题中的应用。

This research project deals with such themes as (1) traveling waves in nonlinear diffusion equations, (2) singular limit of nonlinear diffusion equations and interface motion, (3) blow-up in nonlinear heat equations, (4) mathematical analysis of 3D cellular electrophysiological model. The theme (4) was added after the program began. In theme (1), we studied, among other things, traveling waves colliding with obstacles, the stability of planar waves, and the problem of maximizing traveling wave speeds. In theme (3) we studied, among other things, classification of blow-up in Sobolev superlinear heat equations, and an application the theory of braid groups to the dead-core problem.