兰州大学机构库 >数学与统计学院
格微分方程的行波解和整体解
Alternative TitleTraveling waves and entire solutions for lattice differential equation
史振霞
Thesis Advisor李万同
2012-05-19
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword格动力系统 行波解 整体解 比较原理 存在性 唯一性 稳定性
Abstract格动力系统通常指离散空间上常微分方程的无穷维系统或者是差分方程的无穷维系统。一方面格动力系统来自于实际背景中很多实际问题的数学模型。另一方面,它来自于偏微分方程的空间离散化。因此,对格动力系统的研究具有重要的理论和实际意义。本论文研究了格动力系统的行波解和整体解。首先利用行波解在无穷远处指数衰减性的先验估计来构造合适的上、下解,应用上下解方法、比较原理得出具有对流项的双稳格微分方程在周期介质中的整体解的存在性,进一步建立了整体解的唯一性和稳定性。 其次,考虑了二维格上单稳格动力系统的整体解。通过解一个以\ $-k$ 为初始时刻的初值问题序列并结合比较原理和最大值原理,给出了系统整体解的存在性和解对参数的连续依赖性。 在第四部分,通过结合与空间无关的解和具有不同传播速度和不同传播方向的行波解,给出系统的一个上界和下解,并利用比较原理建立了具有静态阶段的反应扩散系统的整体解。最后,考虑了双稳型非局部积分微分方程的周期行波解。运用基于上下解方法和比较原理的挤压技术证明了行波解的稳定性和唯一性,又由单调动力系统的理论得出了行波解的存在性。
Other AbstractLattice dynamical systems usually refer to infinite systems of ordinary differential equations on discrete space or infinite systems of difference equations. We consider traveling wave solutions and entire solutions of lattice dynamical systems. First, we considers the entire solutions of a lattice reaction-diffusion-convection equation with bistable nonlinearity in periodic media by sub-super solutions method and the comparison principle. Uniqueness and stability of entire solutions are established further. Next, we study the entire solutions of a two-dimensional lattice dynamical system with monostable nonlinearity. We gain the existence of entire solutions and the continuous dependence of entire solutions on the parameters. In the third part, some new entire solutions are constructed. Finally, we consider the periodic traveling wave solutions of a nonlocal integro-differential equation. We prove the existence, stability and the uniqueness of traveling wave solutions .
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225381
Collection数学与统计学院
Recommended Citation
GB/T 7714
史振霞. 格微分方程的行波解和整体解[D]. 兰州. 兰州大学,2012.
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