两类反应扩散模型行波解的指数稳定性 Alternative Title Exponential stability of traveling waves of two kinds of reaction-diffusion models 贾中伟 Thesis Advisor 李万同 2011-05-24 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 传染病模型 格微分方程 行波解 比较原理 加权能量方法 指数稳定性 Abstract 抛物方程或系统的行波解理论是现代数学理论发展最迅速的一个分支. 行波解是方程或系统的一种特殊形式的解, 它的波形关于空间在平移意义下是不会发生改变的. 由于行波解在物理、化学、生物、传染病等领域具有广泛的应用, 所以研究行波解的存在性、稳定性及其它性质具有非常重要的意义. 本文主要考虑了两类反应扩散模型的行波解的稳定性. 第二章考虑了一类具有时滞的单调传染病模型的行波解的稳定性. 在x →-∞ 时初始条件扰动指数衰减到零，但在其他位置可以任意大的条件下，首先利用解析半群理论和抽象泛函微分方程理论建立了R 上对应的Cauchy 问题在加权空间下解的存在性和比较原理. 然后利用加权能量结合比较原理的方法得到了时滞单调传染病模型的单稳波前解在适当的指数加权空间里的稳定性.第三章考虑了二维格上的人口动力学模型的行波解的指数稳定性. 采用与第二章中类似的方法, 即加权能量方法结合比较原理的方法, 建立了行波解的指数渐近稳定性. 这个稳定性结果要求初始条件扰动当x →-∞ 时指数衰减到零，但在其他位置可以任意大. Other Abstract The theory of traveling wave solutions of parabolic differential equations or system is one of the fastest developing areas of modern mathematics. Traveling wave solutions are solutions of special type of differential equations or system, Its wave profile is invariant with respect to space under transition processes. Because it is largely applied in physics、 chemistry、biology、epidemiology and other areas, so the study of the existence、stability and other properties of traveling wave fronts has very important sense. This thesis is concerned with the stability of traveling wave fronts of two kinds of reaction-diffusion models. The second chapter is concerned with the stability of traveling waves fronts of a monostable reaction-diffusion epidemic system with delay. When the initial perturbation around the traveling waves decays exponentially as x →-∞, but can be arbitrarily large in other locations, the existence and comparison theorems of solutions in weighted space of corresponding Cauchy problem are first established for the system on R by appealing to the theories of analytic semigroup and abstract functional differential equations, then the methods of weighted-energy method combining comparison principle are applied to solve the stability of monostable fronts of delayed reaction-diffusion systems with monotonicity in some appropriate exponential weighted space and prove the global exponential stability of monostable fronts under the so-called large initial perturbation. The third chapter is concerned with the stability of traveling waves fronts of a population dynamic model on 2D lattice. The same method, that is, weighted-energy method combining with comparison principle is used to prove the exponential stability of traveling wave fronts of the model. The result of this stability requires that the initial perturbation around the wave is also satisfies decays exponentially as x →-∞, but can be arbitrarily large in other locations. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224905 Collection 数学与统计学院 Recommended CitationGB/T 7714 贾中伟. 两类反应扩散模型行波解的指数稳定性[D]. 兰州. 兰州大学,2011.
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