【作者】 史振霞；

【导师】 李万同；

【作者基本信息】 兰州大学 ， 应用数学， 2012， 博士

【摘要】 格动力系统通常指离散空间上常微分方程的无穷维系统或者是差分方程的无穷维系统(如在D维空间中,由全体整数组成的格zD).一方面格动力系统来自于实际背景中,如在生物学、电路理论、材料学、图像处理及化学反应的很多实际问题的数学模型都可归纳为格动力系统.另一方面,它来自于偏微分方程的空间离散化.因此,对格动力系统的研究具有重要的理论和实际意义.本论文研究了格动力系统的行波解和整体解,其中整体解是指对所有的时间t∈R都有定义的解.首先利用行波解在无穷远处指数衰减性的先验估计来构造合适的上、下解,应用上下解方法、比较原理得出具有对流项的双稳格微分方程在周期介质中的整体解.由于伴随着对流现象、空间的离散化以及周期介质的出现,两列沿着相反方向传播的行波有可能具有不同的传播速度(也就是说,c1≠c1),从而失去了两列波所具有的对称性,这就要求我们构造出不同于以往工作中所出现的上、下解.在存在性的基础上,进一步建立了整体解的唯一性和Lyapunov稳定性.其次,考虑了二维格上单稳格动力系统的整体解.由于空间的离散化,最小波速c*的值依赖于波的传播方向θ.对于来自不同方向θ和θ’的行波解,其最小波速可能满足c*(θ)≠c*(θ’),因此,对于来自两个不同方向且具有不同的传播速度的有限多个行波解,其传播速度可能小于c*(θ)和c*(θ’).当传播速度分别大于、小于或者等于c*(θ)和c*(θ’)时,通过解一个以-k为初始时刻的初值问题序列并结合比较原理和最大值原理,给出了系统整体解的存在性和解对参数的连续依赖性.在种群动力学中,格动力系统被用来描述在空间离散斑块环境下,物种种群的增长和入侵过程.二维格上具有静态阶段的反应扩散系统,它描述了种群个体在迁移和静止这两种状态之间的变换,其中种群中的一部分个体具有迁移性,另一部分个体则总是处于静止状态,并且只有物种处于迁移状态时是可再生的.在第四部分,通过结合与空间无关的解和具有不同传播速度和不同传播方向的行波解,给出系统的一个上界和下解,并利用比较原理建立了具有静态阶段的反应扩散系统的整体解.最后,考虑了双稳型非局部积分微分方程的周期行波解.此类方程是利用卷积算子来描述扩散过程的反应方程,它可以从生物种群、空间生态和传染病等众多研究领域中得到.在双稳的假设条件下,运用基于上下解方法和比较原理的挤压技术证明了行波解的稳定性和唯一性,又由单调动力系统的理论得出了行波解的存在性.更多还原

【Abstract】 Lattice dynamical systems usually refer to infinite systems of ordinary differ-ential equations on discrete space or infinite systems of difference equations (such as the D-dimensional integer lattice ZD). Such systems arise, on the one hand, from practical backgrounds, such as modeling in biology, electrical circuit theory, material science, image processing and chemical kinetics can be induced to lattice dynamical systems. On the other hand, they also arise as the spatial discretization of partial differential equations. Therefore, it is more meaningful and valuable in theory and practice to study such equations. In this thesis, we consider traveling wave solutions and entire solutions of lattice dynamical systems. Here, the entire solutions are defined in the whole space and for all time t∈R.Our thesis firstly considers the entire solutions of a lattice reaction-diffusion-convection equation with bistable nonlinearity in periodic media. Using a priori estimates of the exponential decaying of the traveling wave solution at infinity, we can construct suitable sub-super solutions. Then the existence of entire solutions is obtained via sub-super solutions method and the comparison principle. With the appearing of the convection, the discretization of space and the periodic media, two traveling waves solutions with opposite directions may admit different speeds (that is, C1≠C1), and thus lose their symmetry, which requires us to construct the different sub-super solutions. On the basis of existence, uniqueness and Liapunov stability of entire solutions are established further.Next, we study the entire solutions of a two-dimensional (2-D) lattice dy-namical system with monostable nonlinearity. Due to the discretization of space, c* depends on the directionθof the traveling wave. Traveling waves coming from different directionsθandθmay admit c*(θ)≠c*(θ). Therefore, for a limited number of traveling wave solutions with different propagation speeds from differ-ent directions, the propagation speed may be smaller than c*(θ) and c*(θ). While the propagation speeds are larger than, smaller than or equal to c*(θ) and c*(θ). by solving sequence of Cauchy problems starting at times -k with suitable initial conditions, and using the maximum principle and the comparison principle, we gain the existence of entire solutions and the continuous dependence of entire solutions on the parameters.In the population dynamics, lattice dynamical system is used to describe pro-cesses of the growth and invasion of species in the space of discrete plaque environ-ment. A reaction-diffusion system with a quiescent stage on a 2D spatial lattice for a single-species population with two separate mobile and stationary states, de-scribes a species population of which the individuals alternate between mobile and stationary states, and only the mobile ones reproduce. In the third part, using the comparison principle with appropriate sub-solutions and upper estimates, some new entire solutions are constructed by combining spatially independent solutions and traveling fronts with different wave speeds and directions of propagation.Finally, we consider the periodic traveling wave solutions of a nonlocal integro-differential equation with bistable nonlinearity. This kind of equation describes the diffusion process by the convolution operator. It can arise from the biological populations, ecological and infectious diseases and many other research areas. Under the bistable hypotheses, using the sub- and upper- solutions method, comparison principle and squeezing technique, we prove the stability and the uniqueness of periodic traveling wave solutions with phase shift. Then, we prove the existence of traveling wave solutions by the theory of monotone dynamical systems.更多还原