本文選題：反應(yīng)擴(kuò)散方程 + 自由邊界��； 參考：《上海交通大學(xué)》2015年博士論文

【摘要】：反應(yīng)擴(kuò)散方程或方程組常用于物理、化學(xué)、生態(tài)等學(xué)科中一些實(shí)際問(wèn)題的數(shù)學(xué)建模,其各類(lèi)解的存在性及其動(dòng)力學(xué)性態(tài)一直是微分方程理論研究和應(yīng)用問(wèn)題研究的重要課題.本博士學(xué)位論文主要研究了四類(lèi)非線性反應(yīng)擴(kuò)散模型：受水流作用的水生生物模型、遷移和自然選擇共同作用的基因演化模型、種群入侵演化模型及帶雙自由邊界的燃燒模型,其中前兩個(gè)模型是固定區(qū)域上的反應(yīng)擴(kuò)散問(wèn)題,后兩個(gè)模型是移動(dòng)區(qū)域上的反應(yīng)擴(kuò)散問(wèn)題.借助定性分析、最大值原理、比較原理、譜理論、分支理論和上下解方法等理論工具,探討這些反應(yīng)擴(kuò)散問(wèn)題中的一些參數(shù)如：擴(kuò)散系數(shù)、對(duì)流系數(shù)、區(qū)域度量、初值大小、邊界移動(dòng)參數(shù)等對(duì)系統(tǒng)動(dòng)力學(xué)行為的影響,得到了參數(shù)域的剖分以及不同參數(shù)域內(nèi)相應(yīng)系統(tǒng)的不同動(dòng)力學(xué)性態(tài),揭示了參數(shù)值相應(yīng)的不同環(huán)境對(duì)實(shí)際問(wèn)題產(chǎn)生的本質(zhì)作用.理論上推廣或完善了前人的工作,發(fā)展了處理這類(lèi)問(wèn)題的方法和技巧.應(yīng)用上解釋了一些觀察到的實(shí)際現(xiàn)象,為理解這些實(shí)際問(wèn)題的演化機(jī)制提供了理論依據(jù).具體研究?jī)?nèi)容分為如下四個(gè)部分：生活在溪流或河流中的水生生物,在水流作用下不斷被沖到下游或沖出原有的生態(tài)環(huán)境,從而導(dǎo)致物種數(shù)量下降甚至滅絕,但也有一些物種能在這種環(huán)境里一代一代生存下來(lái),這類(lèi)生物現(xiàn)象常被稱(chēng)為“漂移悖論”(drift paradox),早在1954年就被Muller [123]觀察到.人們一直致力于探索這類(lèi)物種能夠持續(xù)生存的機(jī)制,并試圖通過(guò)數(shù)學(xué)建模幫助理解這一生物現(xiàn)象.2001年,Speirs, Gurney [145]用如下反應(yīng)擴(kuò)散對(duì)流方程描述某一物種在水流作用下的動(dòng)力學(xué)行為其中d|為擴(kuò)散系數(shù),α為水流速度.通過(guò)理論研究對(duì)“漂移悖論”給出一種解釋?zhuān)鹤銐驈?qiáng)的隨機(jī)擴(kuò)散能夠平衡掉水流引起的偏向性運(yùn)動(dòng),從而幫助物種持續(xù)生存.2011年,Vasilyeva,Lutscher[149]研究了相同的方程,但假設(shè)下游滿(mǎn)足Neumann邊界條件,即ux(L,t)=0,并得到類(lèi)似結(jié)論.近來(lái),Lutscher,Lewis,McCauley[111]推導(dǎo)出下游滿(mǎn)足更一般的邊界條件,dux(L,t)-αu(L,t)=-bαu(L,t),其中參數(shù)b表示相對(duì)于水流作用的一種損失率,并且b取不同值可以反映出不同的河流環(huán)境特征,如b=0說(shuō)明下游沒(méi)有損失,對(duì)應(yīng)上游的“無(wú)流”邊界條件；b=1說(shuō)明下游處水流引起百分百損失,對(duì)應(yīng)Neumman邊界條件；b→∞說(shuō)明下游損失很?chē)?yán)重,不利于物種生存,對(duì)應(yīng)Dirichlet邊界條件本論文第二章的第一部分研究了滿(mǎn)足這種一般邊界條件的上述單個(gè)方程.通過(guò)建立臨界區(qū)域長(zhǎng)度和臨界水流速度,我們給出了單個(gè)物種持續(xù)生存的充分必要條件,并發(fā)現(xiàn)b=1/2是該問(wèn)題動(dòng)力學(xué)行為發(fā)生轉(zhuǎn)變的轉(zhuǎn)折點(diǎn).進(jìn)一步,為了探索對(duì)流環(huán)境中的擴(kuò)散機(jī)制,本章第二部分研究了相應(yīng)的一類(lèi)兩個(gè)物種競(jìng)爭(zhēng)模型,并假設(shè)兩個(gè)物種僅擴(kuò)散速度不同.借助譜理論和一些分析技巧,我們得到了0≤b1時(shí)系統(tǒng)的全局動(dòng)力學(xué)性態(tài),結(jié)論表明擴(kuò)散慢的物種一定被取代,增加擴(kuò)散速度更適合自然選擇,推廣了Lou,Lutscher[105]中b=1的結(jié)果；然而b1時(shí),系統(tǒng)的動(dòng)力學(xué)行為變得很不同,特別地,我們說(shuō)明了b3/2時(shí),適中的擴(kuò)散速度有可能被選擇.這里我們指出由于邊界條件中參數(shù)b的引進(jìn),半平凡解的穩(wěn)定性不能通過(guò)文獻(xiàn)[105]中的方法得到,然而本章給出的方法可同樣處理情形b=1.另外,為了得到0≤b1時(shí)系統(tǒng)的全局動(dòng)力學(xué)性態(tài),共存解的不存在性是一個(gè)難點(diǎn),我們發(fā)展了新的方法和數(shù)學(xué)技巧來(lái)克服這一難點(diǎn).這一研究成果已發(fā)表,見(jiàn)附錄二論文1.基因遺傳演化規(guī)律是群體遺傳學(xué)理論研究的核心.自1937年Fisher[56]首次提出用偏微分方程描述在遷移和自然選擇共同作用下基因頻率變化規(guī)律以來(lái),大量非線性反應(yīng)擴(kuò)散方程被用來(lái)描述基因的演化過(guò)程,其中常用的一類(lèi)描述兩種等位基因頻率變化的PDE模型如下([125])d△u和g(x)u(1-u)(1+h-2hu)分別表示遷移和自然選擇作用,h用來(lái)衡量占優(yōu)程度：h1,沒(méi)有占優(yōu),共顯；|h|=1,完全占優(yōu),完全顯性；|h|1,絕對(duì)占優(yōu),超顯性.早期,Fleming [57](1975)和Senn[138](1983)分別用泛函和分支理論研究了|h|1時(shí)該模型非平凡穩(wěn)態(tài)解的存在性和穩(wěn)定性.最近,倪維明教授等人發(fā)表了系列工作[127,110],系統(tǒng)地研究了退化情形h=-1,得到了非平凡穩(wěn)態(tài)解的存在性,穩(wěn)定性,以及極限形態(tài)與平凡解u=0和u=1的關(guān)系,而且他們的研究方法可統(tǒng)一處理|h|≤1的情形.本論文第三章在他們的工作基礎(chǔ)上研究了|h|1的情形.與|h|≤1不同的是此時(shí)系統(tǒng)出現(xiàn)三個(gè)有生物意義的平凡解：u=0,u=1+h/2h和u=1.類(lèi)似于前人工作,我們得到非平凡穩(wěn)態(tài)解的存在性,穩(wěn)定性,以及極限形態(tài)與這三個(gè)平凡解之間的關(guān)系,但也觀察到一些不同現(xiàn)象,如：通過(guò)構(gòu)造一個(gè)具體實(shí)例,我們說(shuō)明當(dāng)遷移很慢,即d很小時(shí),該問(wèn)題有一組非平凡解圍繞u=1+h/2h不斷振蕩,從而沒(méi)有極限形態(tài).進(jìn)一步我們借助譜分析,分支理論和上下解方法,較全面地分析了該系統(tǒng)的動(dòng)力學(xué)性態(tài),結(jié)果表明(?)∮g(x)dx的符號(hào)和遷移速度d的大小對(duì)基因頻率的分布至關(guān)重要,并且當(dāng)h→-1時(shí),我們的結(jié)果與[127,110]相關(guān)結(jié)論一致.這一研究成果已發(fā)表,見(jiàn)附錄二論文3.生物種群入侵問(wèn)題是生態(tài)學(xué)研究中的一個(gè)重要問(wèn)題.2010年,杜一宏教授等人[44,40]用自由邊界問(wèn)題來(lái)描述入侵種群向外擴(kuò)張變化的過(guò)程,提出并系統(tǒng)研究了如下模型其中自由邊界x=h(t)表示擴(kuò)張前沿.文獻(xiàn)[44]假設(shè)空間環(huán)境齊次,即m(x)≡m00,證明了入侵種群滿(mǎn)足二分法：要么成功“擴(kuò)張”(h(t)→∞且u→m0),要么最終“滅絕”(h(t)→h∞∞且u→0),并通過(guò)參數(shù)h0和μ給出了“擴(kuò)張”和“滅絕”發(fā)生的充要條件,另外還刻畫(huà)了自由邊界的漸近速度k0=limt→∞h(t)/t.進(jìn)一步,文獻(xiàn)[40]將上述結(jié)果推廣到高維徑向?qū)ΨQ(chēng)情形,并假設(shè)空間環(huán)境弱異質(zhì),即0m1≤m(r)≤m2,r=|x|,x∈Rn.值得注意的是入侵種群來(lái)到一個(gè)陌生環(huán)境,適合其生長(zhǎng)的區(qū)域{x：m(x)0}和不利其生長(zhǎng)的區(qū)域{x：m(x)0}在實(shí)際中并存(Cantrell, Cosner [20]).基于這種考慮,本論文第四章考慮了一類(lèi)空間環(huán)境強(qiáng)異質(zhì)情形,即m(z)可以改變符號(hào),此時(shí)研究主特征值關(guān)于參數(shù)h0的變化規(guī)律變得十分困難,文獻(xiàn)[40,44]中通過(guò)參數(shù)h0研究種群“擴(kuò)張”或“滅絕”的方法不再適用.我們將擴(kuò)散系數(shù)D視為變化參數(shù),并分析清楚了主特征值關(guān)于參數(shù)D的變化規(guī)律,從而結(jié)合上下解方法給出入侵種群“擴(kuò)張”或“滅絕”的充分條件.這一思路同樣適用于弱異質(zhì)情形,我們得到了入侵種群“擴(kuò)張”或“滅絕”的充要條件,這與[40]中結(jié)論平行.另外,在更一般的假設(shè)條件下,我們得到了自由邊界的漸近速度,推廣了前人的工作.這些理論結(jié)果表明慢擴(kuò)散總是有利于種群入侵成功,而擴(kuò)散較快時(shí),種群能否入侵成功與其初始密度u0和邊界移動(dòng)參數(shù)μ有關(guān).這一研究成果已發(fā)表,見(jiàn)附錄二論文2.爆破現(xiàn)象是燃燒理論研究中的一個(gè)重要課題.上世紀(jì)七八十年代,非線性發(fā)展方程的爆破理論得到迅速發(fā)展,其中一個(gè)被廣泛研究的拋物型燃燒方程如下對(duì)于該方程,已有結(jié)果表明：p≤1+2/n時(shí),所有非平凡解都在有限時(shí)刻爆破,而p1+2/n時(shí),可能出現(xiàn)全局解只要初值充分小,其中1+2*n為Fujita臨界指數(shù)[61,73,86,155].2001年,法國(guó)數(shù)學(xué)家Souplet及其合作者[55,63]在移動(dòng)區(qū)域上研究了上述方程其中自由邊界x=h(t)表示溫度傳播前沿.他們用能量方法給出解爆破的充分條件,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),對(duì)任意p1.這極大地豐富了之前固定區(qū)域上的結(jié)果.受工作[55,63]的啟發(fā),本文第五章試圖將這些結(jié)論推廣到如下帶非局部化反應(yīng)項(xiàng)和雙自由邊界的燃燒模型(事實(shí)上,燃燒理論中非局部化反應(yīng)項(xiàng)有多種類(lèi)型,關(guān)于空間變量積分只是其中一種,詳細(xì)可參考綜述性文章[143].)非局部化反應(yīng)項(xiàng)帶來(lái)一定的研究困難,一方面它使原先的能量方法失效,需要尋找其他方法給出爆破條件；另一方面,由于它含有自由邊界g(t)和h(t),這使得解的一些先驗(yàn)估計(jì)變得復(fù)雜.雙自由邊界提出更直接的困難,因?yàn)椴恢纼蓷l自由邊界是否會(huì)同時(shí)收斂或者同時(shí)發(fā)散.我們將運(yùn)用一些分析技巧來(lái)克服這些困難.結(jié)論表明初值充分大時(shí),解發(fā)生爆破；初值充分小時(shí),全局快解存在；初值適當(dāng)大時(shí),全局慢解存在.這一研究成果已發(fā)表,見(jiàn)附錄二論文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.

【學(xué)位授予單位】：上海交通大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O175

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