| 一类无穷维时滞格微分方程行波解的渐近稳定性 |
| Alternative Title | Asymptotic Stability Of Traveling Waves For Infinite-dimensional Lattice Differential Equations With Delay
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| 柳莉莉 |
| Thesis Advisor | 王智诚
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| 2016-05-18
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 硕士
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| Keyword | 格微分方程
行波解
渐近稳定性
时滞
非局部扩散
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| Abstract | 伴随着自然科学的进步与发展，格微分方程的应用日益普遍。一方面，在现实生活中格微分方程被用来描述具有离散性质的模型，比如在生物种群中出现的斑块现象、材料物理中的晶体生长。另一方面，在实际应用中需要将偏微分方程空间离散化，即形成格微分方程， 比如在计算机中进行数据模拟。人们发现格微分方程具有比连续偏微分方程更复杂的动力学行为，因此研究格微分方程具有重要的理论和现实意义。在本文中，考虑一类具有非局部扩散和非局部时滞的格微分方程非单调行波解的稳定性。首先，总结格微分方程的发展近况并给出本论文问题的简要介绍。其次，当出生函数非单调时，证明非单调行波解的存在性。所用的方法是Schauder 不动点定理和上极限原理。最后，研究非单调行波解的渐近稳定性。首先引入一个合适的变换函数将方程转换到一个新的方程。然后通过标准的迭代法证明新方程局部解的存在性。最后建立非单调行波解的渐近稳定性。采用的方法是转换能量法结合Fourier 变换和非线性Halanay 不等式。 |
| Other Abstract | With the progress and development of natural science, the application of the lattice differential equation is increasingly common.On the one hand, lattice differential equation is used to describe the model with discrete behavior in real life, such as the phenomenon of patches in population, crystal growth in material physics.On the other hand, it is necessary to make spatial discretization of partial differential equation in the practical application, namely the formation of lattice differential equations, such as in a computer data simulation. It is found that lattice differential equation has more complex dynamics behavior than continuous partial differential equation. Thus, it has important theoretical and realistic significance to study the lattice differential equation. In this thesis, we consider the stability of non-monotone traveling waves to a class of lattice differential equation with nonlocal diffusion and nonlocal delay.Firstly, we summarize the recent development of lattice differential equations and give a brief introduction to the problem of this thesis.Secondly, we prove the existence of non-monotone traveling waves when the birth function is non-monotone. Our methods are Schauder fixed point theorem and the upper limit principal.Lastly, we study the asymptotic stability of non-monotone traveling waves. We first introduce a suitable transform function to switch the equation to a new equation. Then we prove the local existence to the solution of this new equation by the standard iteration technique. Finally, we establish the asymptotic stability of the non-monotone traveling waves. The adopted approach is the transformed energy method combining with Fourier transform and nonlinear Halanay's inequality. |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/224444
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
柳莉莉. 一类无穷维时滞格微分方程行波解的渐近稳定性[D]. 兰州. 兰州大学,2016.
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