全空间R?n中反应扩散方程的非平面行波解

对全空间$\mathbb R^N$中反应扩散方程非平面行波解的研究的主要研究结果做了综述性的介绍. 首先介绍本生灯模型作为非平面行波解的一个例子, 进而给出问题的偏微分方程模型, 以及具有鲜明实际背景的点火温度型和双稳态型这2种重要的非线性源. 然后介绍具这2种非线性源的方程非平面行波解的一些定性性质, 包括解的存在唯一性、 单调性、稳定性和水平集的性质等. 接着, 介绍具KPP型非线性源的方程无穷维非平面行波解流形的存在性, 以及解的单调性、稳定性和最小波速的性质等. 最后介绍一些其它相关的研究工作和这个领域内尚未解决的问题.

Nonplanar Travelling Fronts of Reaction-Diffusion Equations in R~N

The problems about the nonplanar travelling fronts of reaction-diffusion equations in $\mathbb R^N$ are proposed by some French researchers who have obtained many important results in recent ten years. Some main results about these issues are reviewed in this paper. Firstly, as an example of nonplanar travelling fronts, the model of Bunsen flames is introduced. The PDE model of this problem with two important nonlinear sources, that is, ignition temperature source and bistable source which have obvious reality background is given accordingly. Then, some qualitative properties of these nonplanar travelling fronts, including the existence, the uniqueness, the monotonicity, the stability and the properties of the level sets of the solutions are reviewed. Next, the results about the equation with KPP type source, including the existence of an infinite-dimensional manifold of nonplanar fronts, the monotonicity, the stability and the properties of minimal propagation speed are introduced. At last, some other relative results in this field are reviewed and then some open questions in this subject are proposed.