高维空间异质环境下反应扩散方程的非平面波

空间周期的反应扩散方程源自于生物学家考虑的在有益区域和有害区域交替出现的环境下的生物入侵问题。该方程上的数学研究引起了许多学者如Berestycki院士等的极大兴趣，并成为热点问题。对于空间周期的反应扩散方程，目前只有一些关于脉冲波的存在性结果，关于非脉冲波解的存在性以及它们的定性性质还有大量的公开问题。与齐次方程的重大区别是脉冲波与传播方向有关，这导致以往使用的利用平面波关于传播方向的不变性、对称性以及移动坐标系等方法失效，使得相关问题的研究变得十分困难。本项目将利用脉冲波关于传播方向的连续可微性，研究空间周期反应扩散方程上的非平面波。主要内容包括：非脉冲波解的存在性证明，它们的定性性质的研究，以及一类水平集随时间变化的整体解的存在性的研究。我们希望通过对空间周期反应扩散方程上非平面波的研究，解决脉冲波关于传播方向不具不变性带来的困难。

Spatially periodic reaction-diffusion equations are first introduced by biologists by considering biological invasions in spatially segmented habitats where favorable and unfavorable zones appear alternately. Mathematical research on such equations has attracted many mathematicians’ interest such as academician Henry Berestycki and become hot topics. For spatially periodic reaction-diffusion equations, there are only a few results about the existence of pulsating fronts and there are many open questions about the existence of non-pulsating fronts and their qualitative properties. The main difference with homogeneous equations is that the pulsating fronts are depending on propagation direction such that we can not use the invariance and symmetry of the profiles of planar fronts with respect to the propagation direction and moving coordinates. This makes corresponding problems very difficult. This project applies the continuity and differentiability of pulsating fronts with respect to propagation direction to study the non-planar fronts of spatially periodic reaction-diffusion equations. Main subjects include: the existence of non-pulsating fronts, their qualitative properties and the existence of an entire solution whose level sets are variant with time variable. We hope to solve the difficulties brought by the dependence of pulsating fronts with respect to the propagation direction through the study of non-pulsating fronts.