本文選題：反應擴散方程 + 行波解��； 參考：《蘭州大學》2017年博士論文

【摘要】：非局部反應擴散方程被認為可以更加準確地描述物理、化學、生態(tài)學中的自然現(xiàn)象,所以受到越來越多的關注.但是隨著非局部時滯的引入,使得原有的許多關于反應擴散方程的研究方法受到了挑戰(zhàn),同時在研究過程中也發(fā)現(xiàn)了許多由非局部時滯作用引起的動力學行為方面的本質變化.目前關于非局部反應擴散方程行波解的研究大都考慮的是非局部時滯充分弱或反應項滿足某些條件,如擬單調、指數(shù)擬單調、弱擬單調以及指數(shù)弱擬單調等等.關于非局部時滯沒有限制時行波解的相關研究很少,而且這些研究結果不能充分揭示非局部反應擴散方程的許多重要性質.另外關于無界區(qū)域上的初值問題以及系統(tǒng)的斑圖生成等問題的研究目前也很少,而這些都是反應擴散方程中的重要問題,因此本文將致力于研究幾類非局部反應擴散方程的行波解、初值問題以及斑圖生成等等.主要內容將分五部分進行闡述.本文首先研究了一類具有Allee效應的非局部反應擴散單種群模型的行波解.由于比較原理不成立,從而基于比較原理的經(jīng)典方法,如上下解方法、移動平面法等都不能應用.因此我們應用Leray-Schauder度理論等方法證得當且僅當波速c≥2r~(1/2)(其中r0是物種的內稟增長率)時,模型存在連接平衡點0到未知正穩(wěn)態(tài)的行波解.進一步利用常數(shù)變易法、柯西-施瓦茲不等式以及一系列分析討論說明了當波速c充分大時,這個未知的正穩(wěn)態(tài)恰好就是方程唯一的正平衡點.此外,針對兩類特殊的核函數(shù),我們還討論了隨著非局部性增強行波解性質的變化,并說明前面所說的未知的正穩(wěn)態(tài)也可能是周期穩(wěn)態(tài).其次研究了帶有聚集項的非局部反應擴散方程的行波解.由于聚集項的出現(xiàn),使模型的解不能被其在零平衡點處的線性化方程所控制.因此,我們借助于一個輔助方程來構造合適的上解,進而證明了連接0到未知正穩(wěn)態(tài)的行波解的存在性.對充分大的波速,我們也證明了未知的正穩(wěn)態(tài)解就是正平衡點.另外,我們還應用上下解方法證明了該模型存在連接0到正平衡點的單調行波.最后,取特殊的核函數(shù),通過數(shù)值模擬的辦法,我們說明隨著非局部性的增強,方程的行波解可能連接0到一個周期穩(wěn)態(tài).進一步借助于穩(wěn)定性分析我們解釋了為什么以及什么時候出現(xiàn)周期穩(wěn)態(tài).第三部分考慮了一類帶有積分項的捕食-食餌模型的初值問題.通過重新定義問題的上下解,并借助于一些輔助函數(shù),我們建立了比較原理,從而構造單調序列并以此給出了初值問題解的存在性和唯一性證明.緊接著借助于輔助方程證明了解的一致有界性.最后,我們給出了初值問題出現(xiàn)Turing分支的條件并通過數(shù)值模擬驗證了這些條件.本文第四部分研究了具有非局部項的Lotka-Volterra競爭系統(tǒng)的行波解.借助于兩點邊值問題和Schauder不動點定理,我們證明了當波速cc*=max{2,2dr~(1/2)}(其中d和r分別是擴散系數(shù)和物種的內稟增長率)時,系統(tǒng)存在連接平衡點(0,0)到未知正穩(wěn)態(tài)的行波解;而當波速cc*時不存在這樣的行波解.最后,針對特殊的核函數(shù),通過數(shù)值模擬的辦法,我們發(fā)現(xiàn)隨著非局部的增強,系統(tǒng)的行波解可能連接平衡點(0,0)到一個周期穩(wěn)態(tài).最后,我們探討了具有非局部項的Lotka-Volterra競爭系統(tǒng)的動力學行為.通過穩(wěn)定性分析,建立了系統(tǒng)出現(xiàn)Turing分支的條件.然后根據(jù)這些條件并結合多尺度分析,得到了關于不同Turing斑圖的振幅方程.接著,通過分析振幅方程的穩(wěn)定性給出了系統(tǒng)出現(xiàn)不同斑圖(包括點狀斑圖和條狀斑圖)的條件.最后,通過數(shù)值模擬結果驗證了我們的理論結果.
[Abstract]:The non local reaction diffusion equation is considered to be more accurate in describing physical, chemical and ecological natural phenomena, so it has attracted more and more attention. However, with the introduction of non local time delay, many existing research methods about the reaction diffusion equation have been challenged, and many of them have been found in the process of research. The essential changes in the dynamic behavior caused by non local delay action. At present, most of the study on the traveling wave solutions of non local reaction diffusion equations is that the nonlocal time delay is fully weak or the reaction term satisfies some conditions, such as quasi monotone, exponential quasi monotone, weakly quasi monotone and weakly quasi monotone, etc. There are few related studies on the time traveling wave solutions, and these results can not fully reveal the many important properties of the nonlocal reaction diffusion equation. In addition, there are few studies on the initial value problem on the unbounded region and the formation of the system speckle patterns. These are all important problems in the inverse diffusion equation. The traveling wave solution, initial value problem and speckle pattern generation of several non local reaction diffusion equations are studied. The main content will be divided into five parts. First, the traveling wave solution of a class of non local reaction diffusion single population model with Allee effect is studied. The method, such as the method of upper and lower solutions and the moving plane method, can not be applied. Therefore, we apply the Leray-Schauder degree theory and other methods to prove that the model has a traveling wave solution that connects the equilibrium point 0 to the unknown steady state when the wave velocity is C > 2r~ (1/2) (and R0 is the intrinsic growth rate of the species). The Cauchy Schwartz inequality is further used by the constant variation method. And a series of analysis and discussion shows that the unknown positive steady state is the only positive equilibrium point of the equation when the wave velocity C is sufficiently large. In addition, we also discuss the changes in the properties of the traveling wave solutions with the non local enhancement for the two class of special kernel functions, and show that the unknown positive steady state mentioned above may also be a periodic steady state. The traveling wave solution of a nonlocal reaction diffusion equation with an aggregation term is studied. Due to the appearance of the aggregation term, the solution of the model can not be controlled by the linearized equation at the zero equilibrium point. Therefore, we construct a suitable upper solution with the aid of an auxiliary equation, and then prove the existence of the traveling wave solution of the 0 to the unknown positive steady state. For the full wave velocity, we also prove that the unknown positive steady state solution is the positive equilibrium point. In addition, we also use the upper and lower solutions to prove that the model has the monotone traveling wave between 0 and the positive equilibrium points. Finally, we take the special kernel function, and through the numerical simulation, we show that the traveling wave solution of the equation with the non local enhancement. We can connect 0 to one periodic homeostasis. Further by the stability analysis we explain why and when the periodic steady state appears. The third part considers the initial value problem of a predator-prey model with integral terms. By redefining the upper and down solutions of the problem and using some auxiliary functions, we have established a comparative original. The existence and uniqueness of the solution of initial value problem are proved by constructing the monotone sequence, and the uniform boundedness of the understanding is proved by the aid of the auxiliary equation. Finally, we give the condition of the Turing bifurcation of the initial value problem and verify these conditions by numerical simulation. The fourth part of this paper studies the non local conditions. With the help of the two point boundary value problem and the Schauder fixed point theorem, we prove that when the wave velocity cc*=max{2,2dr~ (1/2)} (of the D and R is the intrinsic growth rate of the diffusion coefficient and the species), the system has a traveling wave solution that connects the equilibrium point (0,0) to the unknown positive steady state when the wave velocity cc*=max{2,2dr~ (cc*=max{2,2dr~)} (and D and R are respectively); but when the wave velocity cc* does not exist, there is no existence. In the end, by means of numerical simulation, we find that the traveling wave solution of the system may connect the equilibrium point (0,0) to a periodic steady state with the non local enhancement. Finally, we discuss the dynamic behavior of the Lotka-Volterra competitive system with non local terms. The condition of the Turing branch of the system is presented. Then, according to these conditions and combined with multiscale analysis, the amplitude equation about different Turing speckle patterns is obtained. Then, the conditions of the system appearance of different speckle patterns (including spot pattern and bar pattern) are given by analyzing the stability of the amplitude equation. Finally, the numerical simulation results are used to verify that The results of our theory.
【學位授予單位】：蘭州大學
【學位級別】：博士
【學位授予年份】：2017
【分類號】：O175

【參考文獻】

相關期刊論文 前1條

1 ;Travelling Wave Solutions in Delayed Reaction Diffusion Systems with Partial Monotonicity[J];Acta Mathematicae Applicatae Sinica(English Series);2006年02期

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本文編號：2011911