两类时滞非局部扩散系统的行波解

发布时间：2018-06-30 09:15

本文选题：时时滞 + 非非局部扩散　；参考：《兰州交通大学》2017年硕士论文

【摘要】：作为反应扩散方程的一类稳态解,行波解具有空间平移不变性,自然界的许多传播现象都可以用它来描述,例如传染病的传播、种群的增长,物种的迁徙和入侵等.其中,传染病模型中的行波解表示传染源在空间的传播,如果总能将传染源的传播速度控制在恰好发生传染病传播的最小传播速度以内,那么传染病将不会发生.另外,时间滞后和空间的非局部效应都会对方程动力学行为的变化产生影响,例如时滞降低最小波速、非局部扩散加快最小波速等,因此,研究时滞非局部扩散方程行波解的存在性、时滞和非局部扩散对行波解产生的影响,具有重要的现实意义和理论价值.基于此,本文主要研究一类带非局部扩散和非线性发生率的时滞SIR模型和一类非拟单调的时滞非局部扩散系统单稳行波解的存在性以及时滞和非局部扩散对行波解产生的影响.主要工作如下:·研究了一类带非局部扩散和非线性发生率的时滞SIR模型行波解的存在性和非存在性.首先,在一个有限区域内,利用上下解和Schauder不动点定理建立有界区域上的解的存在性;然后通过作先验估计并结合一个极限过程得到全空间上系统行波解的存在性.其次,利用双边Laplace变换建立系统行波解的非存在性.最后,进一步讨论时滞τ和易感者的扩散率对最小波速的影响.·研究了一类非拟单调的时滞非局部扩散系统单稳行波解的存在性.首先利用拟单调条件下单稳波前解的结果,在适当的Banach空间中构造一个拟单调的上下比较系统和波廓集;再利用Schauder不动点定理证明波廓集中的算子的不动点正是非拟单调系统的行波解.
[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.
【学位授予单位】：兰州交通大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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