带自由边界KPP型扩散方程在时间几乎周期介质中的传播现象

发布时间：2018-10-09 08:21

【摘要】：本博士论文研究带自由边界KPP型扩散方程在时间几乎周期介质中解的传播现象。具体来说,我们回顾并探讨了KPP型扩散方程在时间几乎周期介质中解的渐近动力学行为,研究了带自由边界KPP型扩散方程的全局解在时间几乎周期空间非均匀环境中的传播与消亡二分性,进一步证明了在空间均匀环境中传播发生时传播速度由时间几乎周期半波解唯一所决定。论文的具体安排如下：在第一章中,我们简要介绍了经典反应扩散方程及其生态学应用,回顾了自由边界问题的研究现状,总结了本文的研究结果。在第二章中,我们介绍了几乎周期函数、主李雅普诺夫指数、半度量的基本定义和基本性质,回顾了自由边界问题的比较原理以及零点数性质。在第三章中,我们回顾并探讨了KPP型扩散方程在时间几乎周期空间非均匀环境中解的渐近动力学行为。首先对于有界区域上的情形,在其线性化方程主李雅普诺夫指数存在的假设下,我们回顾了几乎周期正解的存在,唯一及稳定性理论。基于该方面的结论,我们探讨了无界区域上该方程几乎周期正解的存在稳定性等相关结论。在第四章中,我们考察带自由边界KPP型扩散方程的全局解在时间几乎周期空间非均匀环境中的传播与消亡二分现象。具体来说,以扩张前沿和扩张能力为参数,我们得到了判断传播与消亡的充分条件。在第五章中,特别地,我们考虑在时间几乎周期空间均匀环境中传播发生时,几乎周期半波解的存在、唯一及稳定性理论,进一步指出了自由边界问题的传播速度与相应波速一致。具体来说,首先,我们讨论了无界区域上带自由边界的KPP方程可以等价转化为无界区域上的固定边界的KPP方程,对这两类方程利用零点数方法和半度量工具给出了一些基本性质。进一步,我们证明了几乎周期正解的存在及稳定性,由等价性从而得到了几乎周期半波解的存在,唯一及稳定性。其次,我们证明了该自由边界问题的传播速度由半波解唯一决定。最后,对于双边自由边界问题我们通过类似的证明方法得到了同样的结论。
[Abstract]:In this paper, we study the propagation of solutions of KPP type diffusion equations with free boundaries in almost periodic media. Specifically, we review and investigate the asymptotic dynamics of solutions of KPP type diffusion equations in time-almost periodic media. The propagation and extinction dichotomy of the global solution of the KPP diffusion equation with free boundary in a time almost periodic space inhomogeneous environment is studied. It is further proved that the propagation velocity is determined only by the almost periodic half-wave solution when propagation takes place in a spatial uniform environment. In the first chapter, we briefly introduce the classical reaction-diffusion equation and its ecological application, review the research status of free boundary problem, and summarize the results of this paper. In the second chapter, we introduce the basic definition and properties of almost periodic function, principal Lyapunov exponent, semi-metric, and review the comparison principle and zero number property of free boundary problem. In chapter 3, we review and discuss the asymptotic dynamics of solutions of KPP type diffusion equations in time almost periodic space inhomogeneous environment. Firstly, for the bounded domain, we review the existence, uniqueness and stability theory of almost periodic positive solutions under the assumption of the existence of the principal Lyapunov exponent of the linearized equation. Based on these conclusions, we discuss the existence and stability of almost periodic positive solutions of the equation in unbounded regions. In chapter 4, we investigate the dichotomy of global solutions of KPP diffusion equations with free boundaries in non-uniform environments in almost periodic space. Specifically, we obtain sufficient conditions for judging propagation and extinction by taking the expansion frontier and the ability of expansion as parameters. In Chapter 5, in particular, we consider the existence, uniqueness and stability theory of almost periodic half-wave solutions when propagation occurs in a time almost periodic space uniform environment. It is further pointed out that the propagation velocity of the free boundary problem is consistent with the corresponding wave velocity. Specifically, first of all, we discuss that the KPP equation with free boundary on the unbounded region can be equivalent to the KPP equation with fixed boundary on the unbounded region. Some basic properties of these two kinds of equations are given by using zero number method and semi-metric method. Furthermore, we prove the existence and stability of almost periodic positive solutions. From the equivalence, we obtain the existence, uniqueness and stability of almost periodic half-wave solutions. Secondly, we prove that the propagation velocity of the free boundary problem is determined only by the half-wave solution. Finally, we obtain the same conclusion for the bilateral free boundary problem by a similar proof method.
【学位授予单位】：中国科学技术大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

【相似文献】