关于趋化模型解的性质研究

发布时间：2018-05-29 06:50

  本文选题：趋化模型 + 整体存在性　； 参考：《电子科技大学》2017年博士论文

【摘要】：生物学、生态学、医学等领域中存在着大量的非线性现象,比如趋化(chemotaxis)现象、趋触(haptotaxis)现象等。为了理解这些现象的复杂形成过程,数学建模与分析已变得愈发重要。由于许多非线性现象都是种群密度分布的外部表现,因而研究种群密度分布已经成为众多学者感兴趣的问题之一。种群密度分布在数学上可通过偏微分方程来刻画,对于这些有着实际背景的偏微分方程解的性质研究已经成为偏微分方程领域的重要课题之一。本文主要对刻画趋化现象的偏微分方程组解的性质进行了研究。研究内容与主要结果如下:1.研究了一类具有非线性扩散和Logistic源项的抛物 椭圆型吸引 排斥趋化模型的初边值问题。该模型刻画了细胞或微生物在化学吸引信号、化学排斥信号、非线性扩散和Logistic源项综合作用下的趋化运动现象。首先,通过不动点定理和抛物、椭圆方程正则理论得到非退化扩散模型经典解的局部存在性和唯一性;其次,利用能量估计的方法得到了当排斥信号强于吸引信号或非线性扩散足够强或Logistic阻尼足够强时,非退化扩散模型经典解的整体存在性和一致有界性;再次,得到了退化扩散模型在相同条件下至少存在一个全局有界的弱解;最后,得到了非退化扩散模型具有一类特殊Logistic源项时的经典解的大时间行为。2.研究了一类二维拟线性抛物 抛物型吸引 排斥趋化模型的初边值问题。由于半线性模型在二维光滑有界域上当吸引信号强于排斥信号时存在有限时间爆破的解,基于能量估计通过考虑非线性扩散得出:在二维光滑有界域上当吸引信号强于排斥信号时,任意超线性扩散都可阻止解的有限时间和无限时间爆破。从而,得出了在非退化扩散情形下该模型存在整体有界的经典解,在退化情形下该模型存在全局有界的弱解。3.研究了一类高维拟线性耗氧趋化模型的初边值问题。不同于上述两类模型,该模型中化学物质(如氧气)是被细菌或微生物消耗。利用化学物质浓度的L∞估计构造了一个新的插值不等式,建立了组合能量估计,得出了该模型在非退化扩散情形下存在整体有界的经典解,在退化扩散情形下该模型存在全局有界的弱解。4.研究了一类具有退化扩散和旋转流的耗氧趋化模型的初边值问题。该模型中趋化敏感函数是个张量函数且它的模满足细胞密度函数的超线性增长性。首先,构造了一个具有非退化扩散和好的边界条件的逼近问题;其次,基于能量估计的方法,得到了该逼近问题整体有界的经典解;最后,通过收敛性分析,得出了全局有界弱解的存在性。
[Abstract]:There are many nonlinear phenomena in biology, ecology and medicine, such as chemotaxisphenomenon and haptotaxisphenomenon. In order to understand the complex forming process of these phenomena, mathematical modeling and analysis has become more and more important. Because many nonlinear phenomena are the external manifestation of population density distribution, the study of population density distribution has become one of the problems of interest to many scholars. Population density distribution can be described mathematically by partial differential equations. The study of the properties of solutions of these partial differential equations with practical background has become one of the important topics in the field of partial differential equations. In this paper, the properties of solutions of partial differential equations which depict chemotaxis are studied. The research contents and main results are as follows: 1. The initial-boundary value problem of a class of parabolic elliptic attractor repulsive chemotaxis model with nonlinear diffusion and Logistic source term is studied. The model describes the chemotaxis of cells or microorganisms under the combined action of chemical attraction signal, chemical rejection signal, nonlinear diffusion and Logistic source term. Firstly, by using fixed point theorem and parabola, the canonical theory of elliptic equation is used to obtain the local existence and uniqueness of the classical solution of nondegenerate diffusion model. Using the method of energy estimation, the global existence and uniform boundedness of the classical solution of the nondegenerate diffusion model are obtained when the repellent signal is stronger than the attractive signal or the nonlinear diffusion or the Logistic damping is strong enough. We obtain at least one globally bounded weak solution for the degenerate diffusion model under the same conditions, and finally, we obtain the large time behavior of the classical solution of the nondegenerate diffusion model with a special Logistic source term. In this paper, the initial-boundary value problem of a class of two-dimensional quasilinear parabolic attractor repulsive chemotaxis model is studied. Because the semilinear model has finite time blow-up solution when the attractive signal is stronger than the repulsive signal in the two-dimensional smooth bounded domain. Based on energy estimation, considering nonlinear diffusion, it is concluded that when the attraction signal is stronger than the repulsive signal in the two-dimensional smooth bounded domain, any superlinear diffusion can prevent the finite time and infinite time explosion of the solution. Thus, the classical solution of the model with global boundedness is obtained in the case of non-degenerate diffusion, and the global bounded weak solution of the model is obtained in the case of degenerate. In this paper, the initial boundary value problem of a class of high dimensional quasilinear oxygen consumption chemotactic model is studied. Unlike these two models, chemicals such as oxygen are consumed by bacteria or microorganisms. A new interpolation inequality is constructed by using the L 鈭，

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