非均匀介质中反应扩散方程的广义行波解 Alternative Title Generalized Traveling Waves of Reaction Diffusion Equations in Heterogeneous Media 舒雅琴 Subtype 博士 Thesis Advisor 李万同 2011-11-26 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 博士 Keyword 反应扩散方程 非均匀介质 上下解 非局部扩散方程 广义行波解 Abstract 众所周知, 自然界中的诸多现象都可以通过反应扩散方程来模拟.在诸如生态学等学科中导出了用积分算子来表示非局部扩散的反应扩散方程.在研究扩散现象时, 时空环境的各向异性(非均匀介质) 是普遍存在的.因此, 研究非均匀介质中的扩散方程具有重要的理论和现实意义.而广义行波解的研究则是其中一个重要分支.这里所谓的广义行波解是定义在所有时间t\in R上的经典行波解在非均匀介质中的一种推广. 首先, 本文研究了非均匀介质中双稳型反应扩散方程的广义行波解. 它连结平衡点 0 和 1.以经典行波解作为初始值来构造初值问题, 通过一系列的估计,证明当初始时刻t=-n 取n→+∞时,相应的初值问题的解序列收敛到某个极限函数便是广义行波解.接着利用上, 下解方法证明了广义行波解的平移唯一性和指数稳定性.此外, 通过借助光滑截断函数来构造一个非均匀介质中的点火型问题,得到了空间非均匀介质中单稳型反应扩散方程广义行波解的存在性. 其次, 讨论了非均匀介质中非局部扩散方程的广义行波解. 非线性项f(x,u) 是完全非均匀的, 从而导致了非平凡正平衡解的出现.利用特征值问题得到了非平凡正平衡解的唯一性.借助比较原理和上下解方法, 证明了连结 0和非平凡正平衡解的广义行波解的存在性. 最后, 研究了依赖于时间的非局部扩散 KPP 方程的广义行波解.通过将退化波方程正则化, 构造一系列的辅助方程, 结合上下解和迭代技巧,得到了正则化后的方程的时间全局解的存在性.再通过标准的极限过程和解的先验估计, 证明了广义行波解的存在性. Other Abstract It seems that many natural phenomena can be modeled by reaction-diffusion equations. Some nonlocal dispersal equations which represented by the integral operator have been derived from the research in many disciplines. Anisotropy of the temporal space environment (heterogeneous media) are always presented in natural diffusion phenomena.Therefore, it is more meaningful and valuable in theory and practice to study such equations in heterogeneous media. One important issue is their generalized traveling waves, which is a global in time solution that can be viewed as a generalization of all usual notions. Firstly, we study the generalized traveling waves of reaction-diffusion equation with bistable nonlinearity in heterogeneous media. The generalized traveling waves connect 0 with 1. By solving a Cauchy problem starting at some time t=-n with the traveling wave as an initial data, then taking the limit n→+∞. The limit function of the sequence of solutions for the Cauchy problem is the generalized traveling waves. Then by constructing subsolution and supersolution coupled , we establish the uniqueness and exponential stability of the generalized traveling waves. In the following, using a smooth cut-off function to construct an ignition problem , we obtain the existence of the generalized traveling waves for the monostable equations in heterogeneous media. Secondly, we investigate the generalized traveling waves of nonlocal dispersal equations in space heterogeneous media, so there exists a nontrivial positive steady state. By the principal eigenvalue problem, the steady state is unique. We apply the comparison principle and sub-super solution method to prove the existence of the generalized traveling waves connecting 0 with the nontrivial positive steady state. Finally, we consider the generalized traveling waves of time dependent nonlocal dispersal KPP equation. By modifying the degenerate parabolic equation as a regular parabolic equation, constructing a series of auxiliary problem coupled with upper and lower solutions and iteration scheme, we prove the existence of the time global solution for the regular parabolic equation. From the limiting procedure and standard estimates, we obtain the existence of the generalized traveling waves. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/225463 Collection 数学与统计学院 Recommended CitationGB/T 7714 舒雅琴. 非均匀介质中反应扩散方程的广义行波解[D]. 兰州. 兰州大学,2011.
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