【摘要】：本博士論文研究帶自由邊界KPP型擴散方程在時間幾乎周期介質(zhì)中解的傳播現(xiàn)象。具體來說,我們回顧并探討了KPP型擴散方程在時間幾乎周期介質(zhì)中解的漸近動力學(xué)行為,研究了帶自由邊界KPP型擴散方程的全局解在時間幾乎周期空間非均勻環(huán)境中的傳播與消亡二分性,進一步證明了在空間均勻環(huán)境中傳播發(fā)生時傳播速度由時間幾乎周期半波解唯一所決定。論文的具體安排如下：在第一章中,我們簡要介紹了經(jīng)典反應(yīng)擴散方程及其生態(tài)學(xué)應(yīng)用,回顧了自由邊界問題的研究現(xiàn)狀,總結(jié)了本文的研究結(jié)果。在第二章中,我們介紹了幾乎周期函數(shù)、主李雅普諾夫指數(shù)、半度量的基本定義和基本性質(zhì),回顧了自由邊界問題的比較原理以及零點數(shù)性質(zhì)。在第三章中,我們回顧并探討了KPP型擴散方程在時間幾乎周期空間非均勻環(huán)境中解的漸近動力學(xué)行為。首先對于有界區(qū)域上的情形,在其線性化方程主李雅普諾夫指數(shù)存在的假設(shè)下,我們回顧了幾乎周期正解的存在,唯一及穩(wěn)定性理論�；谠摲矫娴慕Y(jié)論,我們探討了無界區(qū)域上該方程幾乎周期正解的存在穩(wěn)定性等相關(guān)結(jié)論。在第四章中,我們考察帶自由邊界KPP型擴散方程的全局解在時間幾乎周期空間非均勻環(huán)境中的傳播與消亡二分現(xiàn)象。具體來說,以擴張前沿和擴張能力為參數(shù),我們得到了判斷傳播與消亡的充分條件。在第五章中,特別地,我們考慮在時間幾乎周期空間均勻環(huán)境中傳播發(fā)生時,幾乎周期半波解的存在、唯一及穩(wěn)定性理論,進一步指出了自由邊界問題的傳播速度與相應(yīng)波速一致。具體來說,首先,我們討論了無界區(qū)域上帶自由邊界的KPP方程可以等價轉(zhuǎn)化為無界區(qū)域上的固定邊界的KPP方程,對這兩類方程利用零點數(shù)方法和半度量工具給出了一些基本性質(zhì)。進一步,我們證明了幾乎周期正解的存在及穩(wěn)定性,由等價性從而得到了幾乎周期半波解的存在,唯一及穩(wěn)定性。其次,我們證明了該自由邊界問題的傳播速度由半波解唯一決定。最后,對于雙邊自由邊界問題我們通過類似的證明方法得到了同樣的結(jié)論。
[Abstract]:In this paper, we study the propagation of solutions of KPP type diffusion equations with free boundaries in almost periodic media. Specifically, we review and investigate the asymptotic dynamics of solutions of KPP type diffusion equations in time-almost periodic media. The propagation and extinction dichotomy of the global solution of the KPP diffusion equation with free boundary in a time almost periodic space inhomogeneous environment is studied. It is further proved that the propagation velocity is determined only by the almost periodic half-wave solution when propagation takes place in a spatial uniform environment. In the first chapter, we briefly introduce the classical reaction-diffusion equation and its ecological application, review the research status of free boundary problem, and summarize the results of this paper. In the second chapter, we introduce the basic definition and properties of almost periodic function, principal Lyapunov exponent, semi-metric, and review the comparison principle and zero number property of free boundary problem. In chapter 3, we review and discuss the asymptotic dynamics of solutions of KPP type diffusion equations in time almost periodic space inhomogeneous environment. Firstly, for the bounded domain, we review the existence, uniqueness and stability theory of almost periodic positive solutions under the assumption of the existence of the principal Lyapunov exponent of the linearized equation. Based on these conclusions, we discuss the existence and stability of almost periodic positive solutions of the equation in unbounded regions. In chapter 4, we investigate the dichotomy of global solutions of KPP diffusion equations with free boundaries in non-uniform environments in almost periodic space. Specifically, we obtain sufficient conditions for judging propagation and extinction by taking the expansion frontier and the ability of expansion as parameters. In Chapter 5, in particular, we consider the existence, uniqueness and stability theory of almost periodic half-wave solutions when propagation occurs in a time almost periodic space uniform environment. It is further pointed out that the propagation velocity of the free boundary problem is consistent with the corresponding wave velocity. Specifically, first of all, we discuss that the KPP equation with free boundary on the unbounded region can be equivalent to the KPP equation with fixed boundary on the unbounded region. Some basic properties of these two kinds of equations are given by using zero number method and semi-metric method. Furthermore, we prove the existence and stability of almost periodic positive solutions. From the equivalence, we obtain the existence, uniqueness and stability of almost periodic half-wave solutions. Secondly, we prove that the propagation velocity of the free boundary problem is determined only by the half-wave solution. Finally, we obtain the same conclusion for the bilateral free boundary problem by a similar proof method.
【學(xué)位授予單位】：中國科學(xué)技術(shù)大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2016
【分類號】：O175

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