本文選題：趨化模型 + 整體存在性　； 參考：《電子科技大學》2017年博士論文

【摘要】：生物學、生態(tài)學、醫(yī)學等領域中存在著大量的非線性現(xiàn)象,比如趨化(chemotaxis)現(xiàn)象、趨觸(haptotaxis)現(xiàn)象等。為了理解這些現(xiàn)象的復雜形成過程,數(shù)學建模與分析已變得愈發(fā)重要。由于許多非線性現(xiàn)象都是種群密度分布的外部表現(xiàn),因而研究種群密度分布已經(jīng)成為眾多學者感興趣的問題之一。種群密度分布在數(shù)學上可通過偏微分方程來刻畫,對于這些有著實際背景的偏微分方程解的性質(zhì)研究已經(jīng)成為偏微分方程領域的重要課題之一。本文主要對刻畫趨化現(xiàn)象的偏微分方程組解的性質(zhì)進行了研究。研究內(nèi)容與主要結果如下:1.研究了一類具有非線性擴散和Logistic源項的拋物 橢圓型吸引 排斥趨化模型的初邊值問題。該模型刻畫了細胞或微生物在化學吸引信號、化學排斥信號、非線性擴散和Logistic源項綜合作用下的趨化運動現(xiàn)象。首先,通過不動點定理和拋物、橢圓方程正則理論得到非退化擴散模型經(jīng)典解的局部存在性和唯一性;其次,利用能量估計的方法得到了當排斥信號強于吸引信號或非線性擴散足夠強或Logistic阻尼足夠強時,非退化擴散模型經(jīng)典解的整體存在性和一致有界性;再次,得到了退化擴散模型在相同條件下至少存在一個全局有界的弱解;最后,得到了非退化擴散模型具有一類特殊Logistic源項時的經(jīng)典解的大時間行為。2.研究了一類二維擬線性拋物 拋物型吸引 排斥趨化模型的初邊值問題。由于半線性模型在二維光滑有界域上當吸引信號強于排斥信號時存在有限時間爆破的解,基于能量估計通過考慮非線性擴散得出:在二維光滑有界域上當吸引信號強于排斥信號時,任意超線性擴散都可阻止解的有限時間和無限時間爆破。從而,得出了在非退化擴散情形下該模型存在整體有界的經(jīng)典解,在退化情形下該模型存在全局有界的弱解。3.研究了一類高維擬線性耗氧趨化模型的初邊值問題。不同于上述兩類模型,該模型中化學物質(zhì)(如氧氣)是被細菌或微生物消耗。利用化學物質(zhì)濃度的L∞估計構造了一個新的插值不等式,建立了組合能量估計,得出了該模型在非退化擴散情形下存在整體有界的經(jīng)典解,在退化擴散情形下該模型存在全局有界的弱解。4.研究了一類具有退化擴散和旋轉(zhuǎn)流的耗氧趨化模型的初邊值問題。該模型中趨化敏感函數(shù)是個張量函數(shù)且它的模滿足細胞密度函數(shù)的超線性增長性。首先,構造了一個具有非退化擴散和好的邊界條件的逼近問題;其次,基于能量估計的方法,得到了該逼近問題整體有界的經(jīng)典解;最后,通過收斂性分析,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，

本文編號：1949910