Mathematical Sciences: Semilinear Partial Differential Equations and Their Applications

将在三个方面进行研究：(I)全空间反应扩散方程解的稳定性、有限时间爆破、衰减率和寿命的渐近性；(Ii)空间非均匀非线性热方程解的爆破位置和爆破速率(包括渠道流问题中的模型方程)；(3)半线性椭圆型方程(包括薛定谔方程、趋化性方程和生物物理二元混合问题中的方程)解的集中现象。提出的问题具有数学、物理和生物方面的重要性。该项目中的一些方程多年来一直为数学家所熟知，但关于这些方程的新的、具有挑战性的问题仍然没有得到回答。这些问题的解决不仅将加深我们对这些具体模型的理解，而且还将影响我们对数学、物理和生物领域其他相关问题的理解和解决。

Research will be performed in three areas: (i) stability, finite time blow-up, decay rates, and asymptotics of life span of solutions to reaction-diffusion equations in the whole space; (ii) locations and rates of blow-up of solutions to spatially non-uniform nonlinear heat equations (model equations including the ones arising in the channel flow problem); (iii) concentration phenomena of solutions of semilinear elliptic equations (including Schrodinger equations and the equations arising in chemotaxis and binary mixture problems in biology and physics). The problems proposed are of mathematical, physical and biological importance. Some of the equations in the project have been well-known to mathematicians for years, yet new and challenging questions about these equations remain unanswered. The solutions for these problems will not only deepen our understanding of these concrete models, but also have impact on the understanding of and resolution to other related problems in the fields of mathematics, physics and biology.