两类非线性反应扩散方程解的定性研究

发布时间：2018-08-09 09:18

【摘要】：反应扩散现象普遍存在于自然界,反应扩散方程在现代科学技术中具有重要的应用,它主要研究某个自然系统的空间分布情况与扩散规律,分析时间与空间对系统扩散的影响,从而更准确地把握扩散速率对周围环境造成的影响.最典型的反应扩散模型就是生物学中的捕食-被捕食模型和化学中的Sel'kov模型.研究反应扩散方程有许多种方法,例如上下解方法,拓扑度理论和偏微分定理中的能量估计方法等,运用这些方法研究解的性态可以更好地了解一些非线性反应扩散方程解的性质和行为.本文主要通过分析非常值解或行波解的存在性,不存在性与渐近稳定性,从而对两类非线性反应扩散方程展开定性研究.本文包括如下三章:第一章对反应扩散方程的背景和研究意义,以及捕食-被捕食模型行波解,渐近性,Sel'kov模型和饱和率做了简单介绍,并且介绍了本文的主要工作.第二章研究一个n维扩散的捕食-被捕食系统.通过利用上下解方法和Schauder不动点定理,证明了行波解的存在性与最小波速,运用渐近分析技巧证明了行波解的渐近稳定性,解决了 2014年提出的一个公开问题,将原来一个三维系统问题推广到任意有限维(n维)问题,推广和补充了文献中的相关结论.第三章研究一个具有饱和率的Sel'kov模型.通过运用隐函数定理和Leray-Schauder拓扑度理论,证明非常值正解的稳定性,存在性与不存在性,且提出了一个全新的结论:饱和率的大小影响系统正解的性态,决定Turing图式的形成:较小的饱和系数产生Turing图式,较大的饱和系数则不会产生Turing图式.
[Abstract]:The phenomenon of reaction diffusion exists generally in nature, and reaction diffusion equation has important applications in modern science and technology. It mainly studies the spatial distribution and diffusion law of a natural system, and analyzes the influence of time and space on system diffusion. Thus, the effect of diffusion rate on the surrounding environment is more accurately understood. The most typical reaction-diffusion model is the predator-prey model in biology and the Sel'kov model in chemistry. There are many methods to study the reaction diffusion equation, such as the upper and lower solution method, the topological degree theory and the energy estimation method in the partial differential theorem, etc. Using these methods to study the behavior of solutions can better understand the properties and behavior of solutions of some nonlinear reaction diffusion equations. In this paper, two kinds of nonlinear reaction-diffusion equations are studied qualitatively by analyzing the existence, nonexistence and asymptotic stability of nonconstant solution or traveling wave solution. This paper includes three chapters as follows: in chapter one, the background and significance of the reaction diffusion equation, the traveling wave solution of the predator-prey model, the asymptotic behavior of Selkov model and the saturation rate are briefly introduced, and the main work of this paper is also introduced. In chapter 2, we study an n-dimensional diffusion predator-prey system. By using the upper and lower solution method and Schauder fixed point theorem, the existence and minimum wave velocity of traveling wave solution are proved. The asymptotic stability of traveling wave solution is proved by using asymptotic analysis technique, and an open problem proposed in 2014 is solved. In this paper, the original three-dimensional system problem is extended to any finite dimensional (n-dimensional) problem, and the relevant conclusions in the literature are generalized and supplemented. In chapter 3, we study a Sel'kov model with saturation rate. By using implicit function theorem and Leray-Schauder topological degree theory, we prove the stability, existence and nonexistence of positive solutions of non-constant value. We also propose a new conclusion that the magnitude of saturation ratio affects the behavior of positive solutions of systems. The formation of Turing schemata is determined: the smaller saturation coefficient produces the Turing schema, while the larger saturation coefficient does not produce the Turing schema.
【学位授予单位】：江苏师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

【相似文献】