许多实际问题的数学模型都是发展方程, 如量子力学中的薛定谔方程, 流体力学中的纳维-斯托克斯方程, 空间生态学中的反应扩散方程以及格微分方程. 作为连接数学理论与实际应用的桥梁, 研究发展方程具有重要的现实意义.
由于自然环境随时间和/或空间的变化并不总是均匀的, 因此非均匀介质中发展方程的传播动力学受到越来越多学者的关注. 其中, 渐近传播速度和行波解是发展方程最重要的传播动力学问题之一. 虽然时空非均匀性可以更好地描述环境的复杂性, 这也导致了在非均匀介质中直接应用经典理论的困难. 为了研究非均匀介质中发展方程的波传播动力学, 首先需要把经典行波解的定义作合理地延伸, 从而给出广义行(半)波的概念. 研究非均匀介质中的广义行(半)波的存在性并讨论时间和/或空间的非均匀性对广义行(半)波的波型和波速的影响具有重要意义. 本文研究的主要问题是时间和/或空间非均匀介质中模型的渐近传播速度和广义行(半)波的存在性.
第二章研究了一类时空周期非局部时滞反应扩散模型的传播动力学. 首先证明了该模型时空周期解的存在性和全局吸引性, 接着利用相应线性算子的主特征值,建立了该模型在单调和非单调情形下的渐近传播速度的存在性. 进一步, 我们引入广义半波的概念, 通过构造合适的上下解并且应用渐近传播速度的结果, 证明了在非单调情形下, 存在一个临界波速使得当波速高于临界波速时广义半波存在, 而当
波速低于临界波速时广义半波不再存在. 同时观察到, 非单调情形下模型的渐近传播速度与广义半波的临界波速是一致的. 另外, 单调情形下广义行波的存在/不存在性可直接由非单调情形下的分析得到. 最后, 我们借助数值模拟对所得的理论结果进行了解释说明.
第三章研究了一类时间概周期介质中非局部时滞反应扩散方程的传播动力学. 首先, 分析了相应的线性化方程指数衰减解的性质并刻画出了临界波速. 然后通过构造合适的上下解并利用比较方法, 证明了无论出生率函数是否单调, 只要波速的均值高于临界波速时就存在广义半波. 同时, 我们建立了具有紧支撑初值解的一些渐近传播速度的性质. 最后的讨论表明了本章得到的广义半波的临界速度与某些特殊情形中已知的行波解的最小波速是一致的.
第四章研究了时空非均匀介质中一类格微分方程广义行波的存在性和传播速度区间的一些性质. 通过构造合适的上下解并利用比较原理, 建立了一般时间依赖且空间周期介质中离散 Fisher-KPP 型方程广义行波解临界波速的存在性, 并且证明了只要波速的下均值高于临界波速时就存在广义行波. 此外, 在一些特殊介质如时空周期介质或在时间周期和空间均匀介质中, 我们证明了本节建立的临界速度就
是最小波速. 最后, 在适当的假设下, 我们建立了该格微分方程在一般时间和空间依赖情形下的传播速度区间一些性质和广义行波解的存在性.
Many mathematical models for practical problems are evolution equations, such as the Schrödinger equation in quantum mechanics, the Navier-Stokes equation in fluid mechanics, the reaction-diffusion equation in spatial ecology and lattice differential equation. As a bridge between the mathematical theory and the practical applications, it is of great practical significance to study evolution equations. An increasing attention has been paid to the propagation dynamics of heterogeneous evolution equations since the environment may be not homogeneous over time and/or space. The asymptotic speed of spread and traveling waves is one of the most important propagation dynamical issues in evolution equations. Although the space-time heterogeneity can better describe the complexity of the underlying environment, this also leads to the difficulty of applying the classical theory and concepts directly in heterogeneous media. To study the front propagation dynamics of evolution equations in heterogeneous media, one first needs to properly extend the notion of traveling wave solutions in the classical sense, so as to give the concept of generalized transition waves (semi-waves). It is of great importance to study the existence of generalized transition wave (semi-wave) solutions in heterogeneous media, and to understand the influence of the time and/or space heterogeneity on the
wave profiles and wave speeds of such solutions. The topical question in the current thesis to gain the asymptotic speed of spread and the existence of generalized transition waves (semi-waves) for time and/or space heterogeneous models. Chapter 2 is concerned with the propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay. We first prove the existence and global attractivity of time-space periodic solution for the model, and then by a
family of principal eigenvalues associated with linear operators, we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases. Next, we introduce a notion of generalized transition semi-waves for the model, and then by constructing appropriate upper and lower solutions, and using the results of the asymptotic speed of spread, we show that there is a critical wave speed such that generalized transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed, and generalized transition semiwaves do not exist anymore when their wave speed is less than the critical speed. It turns out that the asymptotic speed of spread coincides with the critical wave speed of generalized transition semi-waves in the non-monotone case. In addition, the existence/non-existence of generalized transition waves in the monotone case can be obtained directly from the analysis in the non-monotone case. Finally, numerical simulations for various cases are carried out to support our theoretical results. In chapter 3, the propagation phenomena of a nonlocal delayed reaction-diffusion equation in time almost periodic media is considered. First, we study properties of exponentially decaying solutions for the corresponding linear problem and characterize the critical wave speed. Then, by constructing appropriate upper and lower solutions and using comparison arguments, we show that no matter the birth rate
function is monotone or not, a generalized transition semi-wave exists as soon as the mean value of wave speed is greater than this critical speed. Meanwhile, some spreading properties for solutions with compact supported initial values are also established. Finally, a brief discussion is given to show that the critical speed of generalized transition semi-waves obtained in the present chapter coincides with the minimum speed of traveling waves observed by others for some special cases. In chapter 4, we study the existence of generalized transition waves and some properties of spreading speed intervals for a lattice differential equation with time and space dependence. Firstly, by constructing appropriate subsolutions and supersolutions and using comparison principal, we show that the existence of the critical speed of generalized transition wave for space periodic and time heterogeneous lattice Fisher-KPP equations, and prove that a generalized transition wave solution exists as soon as the least mean of wave speed is above this critical speed. And the critical speed we construct is proved to be minimal in some particular cases,
such as space-time periodic media or time periodic and space homogeneous media. Finally, under the suitable assumptions, we give some properties of spreading speed intervals and the existence of generalized transition waves for the lattice differential equations in general time and space heterogeneous media.