退化Fisher方程解的渐进性研究

We mainly study the asymptotic behavior of solutions to degenerate Fisher equation, and try to arrive at the the existence of the generallized traveling waves and the description of the level set to the solutions. The degeneracy of the reaction causes the traveling waves with non-critical speeds degenerate algebraicly in infinity, which makes the standard theory of classical reaction-diffusion equations and commonly used methods are no longer applicable, such as the degeneracy of the reaction leads to zero is in the essential spectrum , thus we can not apply the spectral methods direactly. And it also results in the Lyapunov function is not existence. Therefore, the study of this problem becomes much more difficult. According to the feature of degeneracy, we will find new research ideas and research methods and we expect obtain deep results and explain some important phenomena and experimental results.

本项目主要致力于退化Fisher方程在更一般初值条件下解的渐近性研究，试图给出广义行波解的存在性以及解的水平集刻画。由于反应项的退化性导致其非临界波速的行波解在无穷远处以代数率衰减，这使得一些研究经典反应扩散方程行波解的标准理论和常用方法不再适用，如反应项的退化性导致0在本质普中，从而谱方法无法直接利用，再者退化性Lyapunove泛函不存在，因此使得该问题的研究变得更加困难。我们将根据退化性这一特点寻找新的研究思路和研究方法，期望在得到一些深刻完整理论研究结果的同时，解释一些重要自然现象和实验成果。