本文選題：時(shí)時(shí)滯 + 非非局部擴(kuò)散　； 參考：《蘭州交通大學(xué)》2017年碩士論文

【摘要】：作為反應(yīng)擴(kuò)散方程的一類穩(wěn)態(tài)解,行波解具有空間平移不變性,自然界的許多傳播現(xiàn)象都可以用它來描述,例如傳染病的傳播、種群的增長,物種的遷徙和入侵等.其中,傳染病模型中的行波解表示傳染源在空間的傳播,如果總能將傳染源的傳播速度控制在恰好發(fā)生傳染病傳播的最小傳播速度以內(nèi),那么傳染病將不會(huì)發(fā)生.另外,時(shí)間滯后和空間的非局部效應(yīng)都會(huì)對方程動(dòng)力學(xué)行為的變化產(chǎn)生影響,例如時(shí)滯降低最小波速、非局部擴(kuò)散加快最小波速等,因此,研究時(shí)滯非局部擴(kuò)散方程行波解的存在性、時(shí)滯和非局部擴(kuò)散對行波解產(chǎn)生的影響,具有重要的現(xiàn)實(shí)意義和理論價(jià)值.基于此,本文主要研究一類帶非局部擴(kuò)散和非線性發(fā)生率的時(shí)滯SIR模型和一類非擬單調(diào)的時(shí)滯非局部擴(kuò)散系統(tǒng)單穩(wěn)行波解的存在性以及時(shí)滯和非局部擴(kuò)散對行波解產(chǎn)生的影響.主要工作如下:·研究了一類帶非局部擴(kuò)散和非線性發(fā)生率的時(shí)滯SIR模型行波解的存在性和非存在性.首先,在一個(gè)有限區(qū)域內(nèi),利用上下解和Schauder不動(dòng)點(diǎn)定理建立有界區(qū)域上的解的存在性;然后通過作先驗(yàn)估計(jì)并結(jié)合一個(gè)極限過程得到全空間上系統(tǒng)行波解的存在性.其次,利用雙邊Laplace變換建立系統(tǒng)行波解的非存在性.最后,進(jìn)一步討論時(shí)滯τ和易感者的擴(kuò)散率對最小波速的影響.·研究了一類非擬單調(diào)的時(shí)滯非局部擴(kuò)散系統(tǒng)單穩(wěn)行波解的存在性.首先利用擬單調(diào)條件下單穩(wěn)波前解的結(jié)果,在適當(dāng)?shù)腂anach空間中構(gòu)造一個(gè)擬單調(diào)的上下比較系統(tǒng)和波廓集;再利用Schauder不動(dòng)點(diǎn)定理證明波廓集中的算子的不動(dòng)點(diǎn)正是非擬單調(diào)系統(tǒng)的行波解.
[Abstract]:As a kind of steady-state solution of the reaction diffusion equation, the traveling wave solution has the spatial translation invariance, many natural propagation phenomena can be described by it, such as the spread of infectious diseases, population growth, species migration and invasion, etc. The traveling wave solution in the infectious disease model indicates the transmission of the source of infection in space. If the transmission speed of the source of infection can always be kept within the minimum speed of transmission of the infectious disease, then the infectious disease will not occur. In addition, the time delay and the nonlocal effect of space will affect the dynamic behavior of the equation, for example, the delay reduces the minimum wave velocity, the non-local diffusion accelerates the minimum wave velocity, and so on. It is of great practical significance and theoretical value to study the existence of traveling wave solutions for delay nonlocal diffusion equations and the influence of delay and nonlocal diffusion on traveling wave solutions. In this paper, we study the existence of a class of time-delay Sir model with nonlocal diffusion and nonlinear incidence and the existence of a class of non-quasi-monotone time-delay nonlocal diffusion systems, and the effects of delay and nonlocal diffusion on the traveling wave solutions. The main work is as follows: the existence and nonexistence of traveling wave solutions for a class of Sir models with nonlocal diffusion and nonlinear incidence are studied. Firstly, the existence of solutions in a bounded domain is established by using the upper and lower solutions and the Schauder fixed point theorem in a finite domain, and then the existence of the traveling wave solutions in the whole space is obtained by a priori estimate and a limit process. Secondly, the nonexistence of the traveling wave solution of the system is established by using the two-sided Laplace transform. Finally, the effects of delay 蟿 and diffusivity of susceptible persons on the minimum wave velocity are discussed. The existence of a class of unsteady traveling wave solutions for a class of nonquasi monotone delay nonlocal diffusion systems is studied. Firstly, by using the results of the simple stable wavefront solution under quasi-monotone condition, a quasi monotone upper and lower comparison system and wave profile set are constructed in a proper Banach space. Then by using the Schauder fixed point theorem, it is proved that the fixed point of the operator in the wave profile set is the traveling wave solution of the nonquasi monotone system.
【學(xué)位授予單位】：蘭州交通大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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本文編號(hào)：2085784