三类源于趋化性的偏微分方程组解的存在性和稳定性

发布时间：2018-03-29 09:39

本文选题：趋化性　切入点：耦合系统　出处：《吉林大学》2015年博士论文

【摘要】：本文研究源于生物趋化性的抛物-双曲耦合、抛物-抛物耦合及多个抛物方程耦合的偏微分方程组解的存在性和稳定性.全文的内容共有五章.具体如下所示：在第一章里,我们分别从宏观和微观的角度,介绍了源于趋化性的Keller-Segel模型的推导.本章我们仅介绍四种趋化性模型的背景：常微分方程-偏微分方程耦合系统、抛物-抛物耦合系统、两种群的趋化模型以及趋化模型-流体模型耦合.此外,本章还介绍了上述几类趋化性模型关于行波解、爆破、解的全局存在性和吸引子等研究结果.在第二章里,对于常微分方程-偏微分方程耦合的趋化性模型,我们考虑线性趋化性势函数与对数趋化性势函数复合情形,以及对数趋化性势函数的方幂情形,通过Hopf-Cole变换,将常微分方程-偏微分方程耦合系统化为抛物-双曲耦合系统.通过研究双曲守恒组激波解的存在性,证明了原系统行波解的存在性.在第三章里,我们证明了抛物-抛物强耦合的趋化模型在Neumann边值条件下解的全局存在性.该系统具有快扩散和非线性源项,由于该模型特有的非线性交错扩散形式以及强耦合性,给数学研究带来了本质上的困难.我们主要利用能量估计,得到了源于Moser迭代技巧的几个重要引理.在第四章里,我们首先建立了源于趋化性的老年性痴呆症患者脑内老年斑的数学模型,并且画出化学反应网络图.进一步根据线性化方法及非负矩阵和图论相关理论,证明了齐次定态解的存在性,并得到该稳定态解的不稳定的充分条件.在第五章里,我们给出总结以及未来研究的展望.
[Abstract]:In this paper, we study the existence and stability of solutions of parabolic hyperbolic coupling, parabolic parabolic coupling and multiple parabolic equations coupling. There are five chapters in this paper. In this chapter, we only introduce four chemotaxis models: ordinary differential equation-partial differential equation coupling system, parabolic-parabolic coupling system. The chemotaxis model of two species and the coupling of chemotaxis model and fluid model are also introduced in this chapter. In addition, the results of the above mentioned chemotaxis models on traveling wave solution, blasting, global existence of solution and attractor are also introduced. For the chemotaxis model coupled with ordinary differential equation and partial differential equation, we consider the combination of linear chemotactic potential function and logarithmic chemotactic potential function, and the power case of logarithmic chemotactic potential function, by Hopf-Cole transformation. The coupled system of ordinary differential equation and partial differential equation is transformed into a parabolic hyperbolic coupled system. By studying the existence of shock wave solution of hyperbolic conservation system, the existence of traveling wave solution of the original system is proved. In this paper, we prove the global existence of the solution of a strongly coupled parabolic chemotaxis model under the Neumann boundary value condition. The system has fast diffusion and nonlinear source terms, because of its unique nonlinear staggered diffusion form and strong coupling. We mainly use energy estimation to obtain several important Lemma derived from Moser iterative technique. In Chapter 4, We first established a mathematical model of senile plaque in the brain of patients with chemotactic Alzheimer's disease, and drew a chemical reaction network diagram. Further, according to the linearization method and the related theories of non-negative matrix and graph theory, The existence of homogeneous stationary solutions is proved, and the sufficient conditions for the instability of the stable solutions are obtained. In chapter 5, we give a summary and prospects for future research.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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