具有多稳态时间周期反应扩散方程的研究

Research on reaction diffusion equations are widely developed in recent decades. However, up to now, people mainly pay attention to the simple monotone and bistable systems, etc. For the study of the dynamic of the multi-stable states systems is little. (Only H. Matano, P. Polacik and some others did some investigation in recent two years.) In this project, we study the qualitative theory of reaction diffusion equations with multi-stable states, especially, in time periodic environments. We will consider the Cauchy problem and free boundary problem satisfying the Stefan condition. More specifically, we give the exact definition of the time periodic propagating terrace to such problems, which is the generalized definition of the traveling wave and can be seen as a pair of finite traveling wave sequences. And then using the maximum principle and discrete Lyapunov functional an more important result is proved, that is, the existence and uniqueness of time periodic minimal propagating terrace. Furthermore, by applying the methods of the monotone dynamical systems, we make use of this terrace to determine the spreading phenomena. Through carrying out this research, we will show that the spreading feature of the wave of the multi-stable states systems in time periodic environments.

反应扩散方程的研究在近几十年来获得了长足的发展。但是，迄今为止人们关注的大多是单稳态、双稳态等较简单的系统。对于多稳态系统动力学的研究，这方面的工作还很少（只有H. Matano，P. Polacik等人最近两年做过一些工作）。本项目拟系统地研究具有多稳态反应扩散方程的定性理论，特别是在时间周期情形下的问题。我们将考虑Cauchy问题和带Stefan条件的自由边界问题，给出相应问题的时间周期propagating terrace的严格数学定义（它是行波解的推广概念，近似地可看成是多层的行波解），利用极值原理、离散Lyapunov泛函等工具建立最小terrace的存在唯一性理论，并使用单调动力系统的手法以terrace来表征解的传播现象。通过该研究，试图阐明时间周期环境中多稳态系统的波的传播特征。

本项目拟研究一类具有多稳态时间周期反应扩散方程，具体分为相应的Cauchy问题和自由边界问题的terrace以及解的动力学。鉴于一般多稳态系统的复杂性，我们的工作首先证明了一类特殊的多稳态系统时间周期半波解的存在性。考虑到对流项大小与行波解/半波解的存在性有密切关系，我们对项目的研究内容在原有基础上做了补充，研究了带自由边界条件的单稳态反应扩散对流方程时间几乎周期半波解的存在性和带Robin边界条件的反应扩散对流方程Cauchy问题解的动力学。进一步，对于一般的多稳态系统自由边界问题terrace的存在惟一性以及解的动力学，目前该研究正在进行中。