趋化模型的自由边值问题以及分数阶趋化模型的研究

发布时间：2018-09-05 14:21

【摘要】：本文主要讨论了生物数学中的一类偏微分方程组所构成的趋化性系统的解的存在性理论.本文共分为四个部分.第一部分即第一章,我们首先回顾了生物学背景和Keller-Segel模型、自由边值问题的历史,然后总结了一些研究现状,最后基于以上事实给出本文的主要结果.第二部分包括第二、三、四章,分别考虑了抛物-椭圆、完全抛物、抛物-双曲构成的三类趋化模型的自由边值问题.在第二章和第三章中,我们在高维径向对称区域中,引入了自由边界所满足的变系数方程,利用压缩映射原理以及椭圆、抛物方程的估计,分别考虑了抛物-椭圆、完全抛物两类趋化模型解的存在性.在第四章中,我们首先考虑了一维区间的固定边界的抛物-双曲模型,得到了此类趋化模型解的存在性.然后我们引入自由边界所满足的变系数方程,利用压缩映射原理以及抛物、双曲方程的估计,得到了此类自由边值问题解的存在性.第三部分包括第五、六章,分别讨论了分数阶趋化模型在Sobolev空间和Besov空间的存在性.在第五章中,我们依据初始值所在空间的不同,利用压缩映射原理和构造近似解序列等方法,分别考虑了分数阶趋化模型在相应Sobolev空间中解的存在性.在第六章中,我们利用构造近似解序列的方法,得到了分数阶趋化模型在Besov空间中解的存在性.第四部分即第七章,我们对所研究的趋化模型做了一些进一步的研究展望.
[Abstract]:In this paper, we mainly discuss the existence theory of solution of chemotaxis system formed by a class of partial differential equations in biological mathematics. This paper is divided into four parts. In the first chapter, we first review the biological background and Keller-Segel model, the history of free boundary value problems, and then summarize some research status. Finally, based on the above facts, we give the main results of this paper. The second part includes the second, third and fourth chapters. We consider the free boundary value problem of three kinds of chemotaxis models, which are parabolic ellipse, complete parabola and parabolic hyperbolic respectively. In the second and third chapters, we introduce the variable coefficient equation satisfied by the free boundary in the high dimensional radial symmetric region. By using the contraction mapping principle and the estimation of the elliptic and parabolic equations, we consider the parabolic ellipse respectively. Existence of solutions for two classes of completely parabolic chemotactic models. In chapter 4, we first consider the parabolic hyperbolic model with fixed boundary of one-dimensional interval, and obtain the existence of solutions for this kind of chemotaxis model. Then we introduce the variable coefficient equation satisfied by the free boundary and obtain the existence of the solution of the free boundary value problem by using the contraction mapping principle and the estimates of the parabolic and hyperbolic equations. In the third part, we discuss the existence of fractional order chemotaxis model in Sobolev space and Besov space respectively. In chapter 5, according to the difference of initial values, we consider the existence of solutions of fractional order chemotaxis in corresponding Sobolev spaces by means of the contraction mapping principle and the construction of approximate solution sequences. In chapter 6, we obtain the existence of solutions of fractional chemotaxis model in Besov space by using the method of constructing approximate solution sequence. In the fourth part, chapter 7, we make some further research prospects on the chemotaxis model.
【学位授予单位】：武汉大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175.8

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