Mathematical Sciences: Semilinear Partial Differential Equations and Their Applications

将在三个方面进行研究：㈠稳定性， 有限时间爆破，衰减率和寿命的渐近性 反应扩散方程的全空间解; (ii)空间解的位置和爆破速率 非均匀非线性热方程（模型方程包括 在渠道流动问题中出现的问题）;（iii） 半线性椭圆方程解的集中现象 方程（包括薛定谔方程和 在生物学中的趋化性和二元混合问题中出现， 物理学）。 提出的问题是数学，物理和 生物重要性。 项目中的一些方程 多年来一直为数学家所熟知，但新的， 关于这些方程的挑战性问题仍然没有答案。 这些问题的解决不仅会加深我们的 了解这些具体的模型，但也有影响 其他有关问题的认识和解决 数学、物理学和生物学领域。

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.