非局部扩散方程的时空传播

发布时间：2017-12-28 09:28

本文关键词：非局部扩散方程的时空传播　出处：《兰州大学》2017年博士论文　论文类型：学位论文

【摘要】：近些年,用卷积算子描述的非局部扩散方程由于其能描述大范围的扩散现象而受到了广泛的关注和研究,其中对于该类方程的行波解与整解的研究是对其时空传播现象的一个重要研究分支.目前已有的大部分结果都是在核函数对称的假设下进行的,然而,在现实生活中,种群或者个体的扩散由于受到阳光、风力、食物等外界因素的影响而具有方向选择性,这就需要用非对称的核函数来模拟种群的扩散过程.本论文的第一部分主要研究具有非对称核函数的非局部扩散方程的行波解与整解.第二部分考虑一类具有对称核函数的非局部传染病模型的行波解与整解.首先,考虑Fisher-KPP型非对称非局部扩散方程的整解.由于核函数的非对称性影响了最小波速的符号和大小,因此首先利用分析的方法对波速进行合理的分类,进而通过研究两列行波解和空间齐次解之间的耦合作用,构造不同的上下解,再由比较原理和Arzela-Ascoli定理得到方程的几类整解及其定性性质.并且还建立了核函数对称时所得整解的唯一性以及关于波速和平移参数的连续依赖性.其次,研究双稳型非对称非局部扩散方程的行波解的渐近行为和整解.在这种情形,通过限制自变量的取值范围,可以同时考虑单稳波和双稳波之间的相互作用,从而得到合并波类整解的存在性.首先利用Jessen不等式对波速的符号进行比较和分类,再利用双边Laplace变换和Ikehara定理证明行波解的指数渐近行为,然后由上下解方法和比较原理得到整解的存在性及其定性性质.特别地,我们构造了一些核函数和非线性项的具体例子来说明本章所建立的结果是有意义和价值的.再次,研究点火型非对称非局部扩散方程的行波解的渐近行为和整解.由于退化性(即f′(0)=0)的影响,不能再利用双边Laplace变换和Ikehara定理来证明行波解在一端的衰减行为.因此,寻找新的方法.通过构造适当的障碍函数,利用比较原理证明行波解的渐近行为,进而得到整解的存在性及其定性性质.为了说明所得结果的理论意义,我们也给出了一些核函数和非线性项的具体例子.最后,考虑一类具有对称核函数的非局部传染病模型的整解.分别在单稳和双稳型非线性项下考虑该传染病系统整解的类型及其性质,从而说明该传染病从开始传播到持续生存或者消亡有很多种传播方式.我们利用上下解方法以及抽象半群理论得到了几类整解的存在性和性质.但由于非局部算子紧性的缺失以及非局部问题解的低正则性,使得所得整解关于空间变量没有足够好的光滑性,因此进一步利用常微分理论,通过对非线性函数做合理的假设,得到了整解关于空间变量的光滑性.
[Abstract]:In recent years, described by a convolution operator non local diffusion equation can describe the diffusion due to its widespread and has received extensive attention and research, the traveling wave of the equation solution and the study of solutions is an important branch of the spatiotemporal propagation phenomena. Most existing results are in under the assumption of symmetric kernel function, however, in real life, the population or individual diffusion with orientation selectivity due to the effect of external factors affected by sun, wind, and other food, it will need to use the asymmetric kernel function to simulate the diffusion process of the population. Traveling wave non local diffusion equation in the first part of this paper mainly study with asymmetric kernel function solutions and entire solutions. In the second part, considering the traveling wave for a non local epidemic model with symmetric kernel function solutions and entire solutions. First of all, considering Fisher-KPP The whole solution of asymmetric non local diffusion equations. Because of asymmetric kernel function affects the sign and magnitude of the minimum velocity, so first by using the method of the reasonable classification of wave velocity, through studying the coupling effect between the two columns and space traveling wave solutions of homogeneous solutions, construct different upper and lower solutions then several kinds of equations and qualitative properties of entire solutions obtained by the comparison principle and Arzela-Ascoli theorem. We have also established the uniqueness of the symmetric kernel function solutions and on the velocity and displacement parameters of continuous dependence. Secondly, study the asymptotic behavior of traveling wave type bistable asymmetric non local diffusion equations and the solutions in this case, the range limit variables, can consider the interaction between single Wenbo and bistable waves, so as to obtain the entire merger wave solution. Firstly, by using the Jessen inequality of Comparison and classification of velocity symbols are using asymptotic behavior of traveling wave solutions that bilateral Laplace transform and Ikehara theorem, and then by the upper and lower solution method and comparison principle, the existence of solutions and obtain the qualitative properties. In particular, we construct some examples of how the kernel function and non line item to illustrate the results the establishment of this chapter is the significance and value of the research. Thirdly, the asymptotic behavior of traveling wave ignition type asymmetric non local diffusion equations and integral solutions. Due to degenerative (i.e., f '(0) =0) influence, can not use the bilateral Laplace transform and Ikehara theorem to prove the decay behavior of traveling wave solutions in one end. Therefore, searching for new methods. By constructing proper function obstacle, using the comparison principle to prove the asymptotic behavior of traveling waves, and then get the whole existence and qualitative properties. In order to explain the significance of the results of, We also give some examples of how the kernel function and the nonlinear terms. Finally, the whole solution of a class of non local epidemic model with symmetric kernel function is considered. Considering the epidemic system solutions in the type and nature of monostable and bistable nonlinearity respectively, which indicates that the spread of infectious disease from the beginning to survive or die there are many kinds of communication methods. We use the upper and lower solution method and abstract semigroup theory and obtained some kinds of the existence of solutions and properties. But due to the low regularity of the lack of non local compactness and non local solutions, so that the entire solution on the space variable is not good enough the smooth, therefore the further use of ordinary differential theory, by making reasonable assumptions on the nonlinear function, the whole solution smoothness on the space variable is obtained.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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