几类反应扩散问题的定性分析

发布时间：2018-05-07 18:01

本文选题：反应扩散方程 + 自由边界　；参考：《上海交通大学》2015年博士论文

【摘要】：反应扩散方程或方程组常用于物理、化学、生态等学科中一些实际问题的数学建模,其各类解的存在性及其动力学性态一直是微分方程理论研究和应用问题研究的重要课题.本博士学位论文主要研究了四类非线性反应扩散模型：受水流作用的水生生物模型、迁移和自然选择共同作用的基因演化模型、种群入侵演化模型及带双自由边界的燃烧模型,其中前两个模型是固定区域上的反应扩散问题,后两个模型是移动区域上的反应扩散问题.借助定性分析、最大值原理、比较原理、谱理论、分支理论和上下解方法等理论工具,探讨这些反应扩散问题中的一些参数如：扩散系数、对流系数、区域度量、初值大小、边界移动参数等对系统动力学行为的影响,得到了参数域的剖分以及不同参数域内相应系统的不同动力学性态,揭示了参数值相应的不同环境对实际问题产生的本质作用.理论上推广或完善了前人的工作,发展了处理这类问题的方法和技巧.应用上解释了一些观察到的实际现象,为理解这些实际问题的演化机制提供了理论依据.具体研究内容分为如下四个部分：生活在溪流或河流中的水生生物,在水流作用下不断被冲到下游或冲出原有的生态环境,从而导致物种数量下降甚至灭绝,但也有一些物种能在这种环境里一代一代生存下来,这类生物现象常被称为“漂移悖论”(drift paradox),早在1954年就被Muller [123]观察到.人们一直致力于探索这类物种能够持续生存的机制,并试图通过数学建模帮助理解这一生物现象.2001年,Speirs, Gurney [145]用如下反应扩散对流方程描述某一物种在水流作用下的动力学行为其中d|为扩散系数,α为水流速度.通过理论研究对“漂移悖论”给出一种解释：足够强的随机扩散能够平衡掉水流引起的偏向性运动,从而帮助物种持续生存.2011年,Vasilyeva,Lutscher[149]研究了相同的方程,但假设下游满足Neumann边界条件,即ux(L,t)=0,并得到类似结论.近来,Lutscher,Lewis,McCauley[111]推导出下游满足更一般的边界条件,dux(L,t)-αu(L,t)=-bαu(L,t),其中参数b表示相对于水流作用的一种损失率,并且b取不同值可以反映出不同的河流环境特征,如b=0说明下游没有损失,对应上游的“无流”边界条件；b=1说明下游处水流引起百分百损失,对应Neumman边界条件；b→∞说明下游损失很严重,不利于物种生存,对应Dirichlet边界条件本论文第二章的第一部分研究了满足这种一般边界条件的上述单个方程.通过建立临界区域长度和临界水流速度,我们给出了单个物种持续生存的充分必要条件,并发现b=1/2是该问题动力学行为发生转变的转折点.进一步,为了探索对流环境中的扩散机制,本章第二部分研究了相应的一类两个物种竞争模型,并假设两个物种仅扩散速度不同.借助谱理论和一些分析技巧,我们得到了0≤b1时系统的全局动力学性态,结论表明扩散慢的物种一定被取代,增加扩散速度更适合自然选择,推广了Lou,Lutscher[105]中b=1的结果；然而b1时,系统的动力学行为变得很不同,特别地,我们说明了b3/2时,适中的扩散速度有可能被选择.这里我们指出由于边界条件中参数b的引进,半平凡解的稳定性不能通过文献[105]中的方法得到,然而本章给出的方法可同样处理情形b=1.另外,为了得到0≤b1时系统的全局动力学性态,共存解的不存在性是一个难点,我们发展了新的方法和数学技巧来克服这一难点.这一研究成果已发表,见附录二论文1.基因遗传演化规律是群体遗传学理论研究的核心.自1937年Fisher[56]首次提出用偏微分方程描述在迁移和自然选择共同作用下基因频率变化规律以来,大量非线性反应扩散方程被用来描述基因的演化过程,其中常用的一类描述两种等位基因频率变化的PDE模型如下([125])d△u和g(x)u(1-u)(1+h-2hu)分别表示迁移和自然选择作用,h用来衡量占优程度：h1,没有占优,共显；|h|=1,完全占优,完全显性；|h|1,绝对占优,超显性.早期,Fleming [57](1975)和Senn[138](1983)分别用泛函和分支理论研究了|h|1时该模型非平凡稳态解的存在性和稳定性.最近,倪维明教授等人发表了系列工作[127,110],系统地研究了退化情形h=-1,得到了非平凡稳态解的存在性,稳定性,以及极限形态与平凡解u=0和u=1的关系,而且他们的研究方法可统一处理|h|≤1的情形.本论文第三章在他们的工作基础上研究了|h|1的情形.与|h|≤1不同的是此时系统出现三个有生物意义的平凡解：u=0,u=1+h/2h和u=1.类似于前人工作,我们得到非平凡稳态解的存在性,稳定性,以及极限形态与这三个平凡解之间的关系,但也观察到一些不同现象,如：通过构造一个具体实例,我们说明当迁移很慢,即d很小时,该问题有一组非平凡解围绕u=1+h/2h不断振荡,从而没有极限形态.进一步我们借助谱分析,分支理论和上下解方法,较全面地分析了该系统的动力学性态,结果表明(?)∮g(x)dx的符号和迁移速度d的大小对基因频率的分布至关重要,并且当h→-1时,我们的结果与[127,110]相关结论一致.这一研究成果已发表,见附录二论文3.生物种群入侵问题是生态学研究中的一个重要问题.2010年,杜一宏教授等人[44,40]用自由边界问题来描述入侵种群向外扩张变化的过程,提出并系统研究了如下模型其中自由边界x=h(t)表示扩张前沿.文献[44]假设空间环境齐次,即m(x)≡m00,证明了入侵种群满足二分法：要么成功“扩张”(h(t)→∞且u→m0),要么最终“灭绝”(h(t)→h∞∞且u→0),并通过参数h0和μ给出了“扩张”和“灭绝”发生的充要条件,另外还刻画了自由边界的渐近速度k0=limt→∞h(t)/t.进一步,文献[40]将上述结果推广到高维径向对称情形,并假设空间环境弱异质,即0m1≤m(r)≤m2,r=|x|,x∈Rn.值得注意的是入侵种群来到一个陌生环境,适合其生长的区域{x：m(x)0}和不利其生长的区域{x：m(x)0}在实际中并存(Cantrell, Cosner [20]).基于这种考虑,本论文第四章考虑了一类空间环境强异质情形,即m(z)可以改变符号,此时研究主特征值关于参数h0的变化规律变得十分困难,文献[40,44]中通过参数h0研究种群“扩张”或“灭绝”的方法不再适用.我们将扩散系数D视为变化参数,并分析清楚了主特征值关于参数D的变化规律,从而结合上下解方法给出入侵种群“扩张”或“灭绝”的充分条件.这一思路同样适用于弱异质情形,我们得到了入侵种群“扩张”或“灭绝”的充要条件,这与[40]中结论平行.另外,在更一般的假设条件下,我们得到了自由边界的渐近速度,推广了前人的工作.这些理论结果表明慢扩散总是有利于种群入侵成功,而扩散较快时,种群能否入侵成功与其初始密度u0和边界移动参数μ有关.这一研究成果已发表,见附录二论文2.爆破现象是燃烧理论研究中的一个重要课题.上世纪七八十年代,非线性发展方程的爆破理论得到迅速发展,其中一个被广泛研究的抛物型燃烧方程如下对于该方程,已有结果表明：p≤1+2/n时,所有非平凡解都在有限时刻爆破,而p1+2/n时,可能出现全局解只要初值充分小,其中1+2*n为Fujita临界指数[61,73,86,155].2001年,法国数学家Souplet及其合作者[55,63]在移动区域上研究了上述方程其中自由边界x=h(t)表示温度传播前沿.他们用能量方法给出解爆破的充分条件,并首次得到全局快解(h(t)→h∞∞且u→ 0)和全局慢解(h(t)→∞且u→ 0),对任意p1.这极大地丰富了之前固定区域上的结果.受工作[55,63]的启发,本文第五章试图将这些结论推广到如下带非局部化反应项和双自由边界的燃烧模型(事实上,燃烧理论中非局部化反应项有多种类型,关于空间变量积分只是其中一种,详细可参考综述性文章[143].)非局部化反应项带来一定的研究困难,一方面它使原先的能量方法失效,需要寻找其他方法给出爆破条件；另一方面,由于它含有自由边界g(t)和h(t),这使得解的一些先验估计变得复杂.双自由边界提出更直接的困难,因为不知道两条自由边界是否会同时收敛或者同时发散.我们将运用一些分析技巧来克服这些困难.结论表明初值充分大时,解发生爆破；初值充分小时,全局快解存在；初值适当大时,全局慢解存在.这一研究成果已发表,见附录二论文5.
[Abstract]:The reaction diffusion equation or equation group is commonly used in mathematical modeling of some practical problems in physics, chemistry, ecology and other disciplines. The existence and dynamic state of all kinds of solutions have been an important subject in the research and application of the theoretical research and application of differential equations. This doctoral dissertation mainly studies four kinds of nonlinear reaction diffusion models: water receiving. The aquatic biological model of flow, the genetic model of migration and natural selection, the model of population invasion and evolution and the combustion model with double free boundary, the first two models are the problem of reaction diffusion on the fixed area, the last two models are the inverse diffusion problems on the moving area. Some theoretical tools such as comparison principle, spectrum theory, branch theory and upper and lower solution method are used to discuss the parameters such as diffusion coefficient, convection coefficient, region measure, initial value size, boundary movement parameter and so on. The parameter domain is dissecting and the corresponding system in different parameter domains is not. The same dynamic state reveals the essential effect of the different environment on the actual problems. In theory, it popularized or perfected the work of the predecessors and developed the methods and techniques to deal with these problems. The practical phenomena were explained in application, and the theoretical basis for understanding the evolutionary mechanism of these practical problems was provided. The specific research contents are divided into four parts: aquatic organisms living in streams or rivers, which are constantly being washed down downstream or out of the original ecological environment under the action of water, resulting in the decline and even extinction of species, but some species can also survive in this environment for generations. This kind of biological phenomenon is often called as a phenomenon. The "drift paradox" (drift paradox) was observed by Muller [123] in 1954. People have been trying to explore the mechanism of this species to survive, and try to help understand this biological phenomenon by mathematical modeling, Speirs, Gurney [145] describes a species in the flow of a species using the following reaction diffusion convection equation. In the dynamic behavior, d| is the diffusion coefficient and the alpha is water velocity. Through theoretical study, the "drift paradox" is explained by the theory that strong enough random diffusion can balance the biased movement caused by the flow of water, thus helping the species to survive for.2011 years, Vasilyeva and Lutscher [149] study the same equation, but assume that the downstream satisfies N. Eumann boundary conditions, that is, UX (L, t) =0, and get similar conclusions. Recently, Lutscher, Lewis, McCauley[111] have deduced that the downstream satisfies the more general boundary conditions, Dux (L, t) - alpha u, which represents a loss ratio relative to the flow of water, and takes different values to reflect the different river environment characteristics, such as the description There is no loss in the downstream, which corresponds to the "no flow" boundary condition in the upstream. B=1 shows that the downstream flow causes 100% loss and corresponds to the Neumman boundary condition. B - Infinity indicates that the downstream loss is very serious and is not conducive to the survival of the species. The first part of the second chapter of this paper corresponding to the boundary condition of Dirichlet is studied to meet the general boundary conditions. By establishing the critical region length and the critical flow velocity, we give the sufficient and necessary conditions for the survival of a single species, and find that b=1/2 is the turning point of the transformation of the dynamic behavior of this problem. Further, in order to explore the diffusion mechanism in the convection environment, the second part of this chapter studies a corresponding class of two objects. It is assumed that the two species only have different diffusion velocity. With the help of spectral theory and some analytical techniques, we get the global dynamic state of the system with 0 < B1. The conclusion shows that the species with slow diffusion must be replaced, the increase of diffusion speed is more suitable for natural selection, and the result of b=1 in Lutscher[105] is extended; however, B1, system We point out that the stability of the semi trivial solution in the boundary condition can not be obtained by the method in the literature [105], but the method given in this chapter can also deal with the case b=1. in addition to the case b=1., in order to obtain the b3/2. The global dynamic state of the system and the non existence of coexisting solutions to 0 or less B1 is a difficult point. We have developed new methods and mathematical techniques to overcome this difficulty. The results of this study have been published. The genetic evolution of 1. genes in Appendix two is the core of the study of the theory of population genetics. Since 1937, Fisher[56] was first proposed to use bias. A large number of nonlinear reaction diffusion equations have been used to describe the evolution process of genes, in which the differential equations describe the variation of gene frequency under the common action of migration and natural selection. One of the commonly used PDE models describing the frequency changes of two alleles ([125]) d delta u and G (x) U (1-u) (1+h-2hu) represent migration and natural selection, respectively. H is used to measure the degree of dominance: H1, no dominant, CO explicit; |h|=1, fully dominant, fully dominant; |h|1, absolute dominance, overdominance. Early, Fleming [57] (1975) and Senn[138] (1983) studied the existence and stability of the non ordinary steady solution of the model with functional and branching theory respectively. Recently, Professor Ni Weiming and others In the series of work [127110], we systematically study the degenerate case h=-1, and obtain the existence, stability, and the relation between the nontrivial steady state solution and the trivial solution u=0 and u=1, and their research methods can deal with the case of |h| less than 1. The third chapter of this paper studies the situation of |h|1 on the basis of his work. And |h| < 1. The difference is that there are three biologically trivial solutions of the system at this time: u=0, u=1+h/2h and u=1. are similar to the previous work. We get the existence, stability, and the relationship between the limit form and the three ordinary solutions, but we also observe some different images, such as: by constructing a concrete example, we explain When the migration is very slow, that is, D is very small, the problem has a group of non trivial solutions that oscillates around u=1+h/2h, thus there is no limit form. Further we analyze the dynamic state of the system in a more comprehensive way by means of spectral analysis, branch theory and upper and lower solutions. The result shows that the size of the symbol and migration speed D of G (x) DX is the frequency of gene frequency. Distribution is very important, and when h to -1, our results are consistent with the [127110] related conclusions. The results of this study have been published. See Appendix two the 3. biological population invasion is an important problem in the ecological study, and Professor Du Yihong et al. [44,40] describes the outward expansion of the invasive population by the free boundary problem. In the process, we propose and systematically study the following model in which the free boundary x=h (T) represents the extension frontier. The document [44] assumes that the spatial environment is homogeneous, that is, m (x) M00, which proves that the invasive population satisfies the dichotomy, or the successful "expansion" (H (T), infinity and U / M0), or the final "extinction" (H (T), infinity and 0), and is given by parameter and mu. The sufficient and necessary conditions for the occurrence of "expansion" and "extinction" are also given. In addition, the asymptotic velocity of the free boundary is also portrayed by k0=limt to h (T) /t.. In literature [40], the above results are extended to the high dimensional radial symmetry, and the weak heterogeneity in the space environment, that is, 0m1 < m (R) < m2, R =|x|, and R =|x|, is worth noting that the invasive population comes to a stranger. The environment, the region {x:m (x) 0} suitable for its growth and the area {x:m (x) 0} that are unfavorable for its growth (Cantrell, Cosner [20]). Based on this consideration, the fourth chapter of this paper considers a class of strong heterogeneity in space environment, that is, m (z) can change the symbol, and it is very difficult to study the change law of the main eigenvalue on the parameter H0. In literature [40,44], the method of studying population "dilatation" or "extinction" by parameter H0 is no longer applicable. We consider the diffusion coefficient D as a change parameter and analyze the change rule of the main eigenvalue on the parameter D, so as to combine the upper and lower solutions to the sufficient conditions for "expansion" or "extinction" of the invading population. In the case of weak heterogeneity, we get the necessary and sufficient condition for the "expansion" or "extinction" of the invasive population, which is parallel to the conclusion in [40]. In addition, under the more general hypothesis, we get the asymptotic velocity of the free boundary, which generalizes the work of the predecessors. These results show that the slow diffusion is always beneficial to the success of the population invasion, When the diffusion is fast, the success of the population is related to the initial density U0 and the boundary movement parameters. The research results have been published. See the 2. blasting phenomenon in Appendix two is an important subject in the theory of combustion. In the 70s and 80s of last century, the explosion theory of nonlinear development equation developed rapidly, one of which was widely used. The parabolic combustion equation is studied as follows. The results show that when p < 1+2/n, all nontrivial solutions are blasting at finite time. While p1+2/n, the global solution may appear as long as the initial value is small, and 1+2*n is the Fujita critical exponent [61,73,86155].2001 year, and the law mathematician Souplet and its collaborator [55,63] are in the mobile area. The above equations are studied in which free boundary x=h (T) represents the front of temperature propagation. They use the energy method to give the sufficient conditions for the solution of blasting, and the global fast solution (H (T), H infinity and u 0) and the global slow solution (H (T), infinity and u 0) are obtained for the first time, and the results of any p1. which are extremely rich in the previous fixed area are greatly enriched. The fifth chapter tries to generalize these conclusions to the following combustion models with non localized reaction terms and double free boundary (in fact, there are many types of non localized reactions in combustion theory, the integral of spatial variables is only one, and the detailed reference [143].) On the one hand, it makes the original energy method invalid and needs to find other methods to give blasting conditions; on the other hand, it contains the free boundary g (T) and H (T), which makes some prior estimates of the solution complex. The double free boundary is more direct because it does not know whether two free boundaries will converge or same at the same time. We will use some analytical techniques to overcome these difficulties. The conclusion shows that the solution occurs when the initial value is sufficiently large, the initial value is full hours, the global fast solution exists, and the global slow solution exists when the initial value is appropriate. This research results have been published, see Appendix two paper 5.

【学位授予单位】：上海交通大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

【相似文献】