数学与统计学院知识库

自从上世纪七十年代以来, 抛物型方程的行波解理论得到了充分的发展. 人们发现行波解能够很好的描述自然界中的振荡现象及有限速度传播问题, 所以它的存在性、唯一性和稳定性等被广泛研究. 在自然界中, 时间滞后总是存在的, 所以自上世纪七十年代以来, 有大量的工作从动力系统和半群的观点出发对偏泛函微分方程（特别是时滞反应扩散方程）进行了研究. 然而, 自上世纪九十年代初以来, 人们在研究中发现时滞反应扩散方程并不能准确地描述某些研究对象的时空行为. 例如对一个生物种群来说, 它在空间上是走动的, 所以它的位置随时间而变化, 这就导致了空间上的非局部交错作用的发生. 受此思想的促动, 在生物种群、空间生态和流行病等领域, 许多具有空间非局部作用的时滞扩散系统（反应扩散方程, 格微分方程）被建立, 试图更为准确地描述研究对象的时空模式. 这类系统一般表现为在非线性项中结合了对过去时间和整个空间的加权平均, 因此称此类系统为非局部时滞反应扩散系统. 尽管这种系统能够更加准确地反映所研究的实际问题, 但时间时滞和空间非局部性也同时导致了数学理论研究上的困难, 并引起了许多动力学行为上的本质改变. 例如, 时滞和非局部性可以导致方程不满足比较原理, 使得解半流很难向后延拓. 时滞可以引起平衡点稳定性的改变, 导致振动发生, 产生混沌等. 就行波解来说, 时滞能够使最小波速降低, 失去单调性, 产生振动以及周期行波解等. 特别,随着非局部项的出现, 解半流通常不再是紧的. 因此, 对此类问题的研究既有重要的理论和实际意义, 也在数学上具有挑战意义的困难. 本论文继续从动力系统的角度研究此类方程, 力图对此类方程发展一些新的研究方法, 建立一些新的抽象结果, 结合具体模型探讨时滞和空间非局部性特别是空间非局部性对方程动力学行为的影响, 发现一些实际问题的未被发现的性质. 本论文主要从行波解和整体解两个方面来进行研究.

本文首先研究了非局部时滞反应扩散系统的波前解的存在性. 通过引入多种单调性条件、G- 紧性和 M- 连续性等概念, 对一类抽象的二阶混合型泛函微分方程建立了解的存在性. 方法是单调迭代结合上下解技术，并且运用非标准序发展了新的迭代序列. 然后运用这些结果到相应的行波系统, 从而建立了波前解的存在性. 这使得关于波前解的存在性结果可以应用到具有各种核函数的非局部时滞反应扩散系统. 作为应用, 详细讨论了一类具有非局部时滞的单物种扩散模型的单调波前解的存在性, 并通过选择几类不同的核函数而获得了一些波前解的存在性准则.

其次, 研究了具有非局部时滞的拟单调反应对流扩散方程的波前解的存在性、唯一性和渐近稳定性. 对非线性项分别考虑了两种情形, 即单稳型和双稳型. 对具有单稳型非线性项的方程, 波前解的存在性是利用前面建立的结果获得的, 而渐近稳定性是利用挤压技术结合比较定理被证明的. 进一步, 证明了波前解在负无穷远处的先验渐近行为, 同时也得到了波前解的不存在性, 进而由稳定性结果直接得到了唯一性. 特别地, 我们发现时间时滞能够降低渐近传播速度而空间非局部作用能够加快渐近传播速度. 对具有双稳型非线性项的方程, 构造了许多对不同的上下解并利用比较原理结合挤压技术证明了方程有唯一的波前解（波速也是唯一的）, 该波前解是单调递增的并且是全局渐近稳定的. 对这两种类型的方程, 都考虑了对流对传播速度的影响. 当应用到一些生物种群和传染病模型中时, 得到了许多有实际意义的结果.

最后, 利用前面得到的单调行波解并结合比较定理, 对具有非局部时滞的拟单调反应扩散方程证明了整体解的存在性. 整体解存在性证明的一个重要思想就是构造适当的上下解来确定整体解在时间变量趋于负无穷大时的渐近行为. 对双稳型方程, 进一步证明了它的唯一性和 Lyapunov 稳定性. 对单稳型方程, 通过考虑行波解和与空间变量无关的整体解的组合, 得到了另外一些新型的整体解. 另外, 我们也利用单调动力系统理论给出了一些判断与空间变量无关的整体解存在的充分条件. 作为主要结果的应用, 分别讨论了两类具有双稳型非线性项和单稳型非线性项的非局部时滞反应扩散模型的整体解.

Since the 1970s, there have been intensive developments in the theory of traveling wave solutions of parabolic differential equations. It was found that traveling waves can well model the oscillatory phenomenon and the propagation with finite speed of nature,so the existence, uniqueness and stability of traveling wave solutions have been widely studied. Due to time delays which usually exist in nature, there have been a number of works devoted to the study of partial functional differential equations (especially, delayed reaction-diffusion equations) from the dynamical systems and semigroups point of view since the 1970s. However, it has become recognized that the delayed reaction-diffusion equations can not accurately describe the spatial-temporal patterns of the objects of study.
For example, consider a biological population. Since individuals are moving around,they have not been at the same point in space at previous times, which results in the
spatial nonlocality. Since then, many delayed diffusion systems with nonlocal effects(reaction-diffusion equations, lattice differential equations) have been proposed to bring
the models closer to the biological reality in population biology, spatial ecology and disease spread. Since such systems generally have the nonlinearities involving a weighted spatial averaging over the whole of the infinite spatial domain and the whole of the previous times, we call them reaction-diffusion systems with nonlocal delays. Though the systems are closer to the reality, the time delay and spatial nonlocality lead to many mathematical difficulties and essential changes of dynamics. For example, comparison theorems are not, in general, applicable for reaction-diffusion equations with nonlocal delays and the backward continuation of solution semiflows is very difficult. The time delay can lead to the change of stability of equilibrium and result in oscillations and chaos. The time delay can also slow the minimal wave speed of traveling wave fronts, make the traveling wave fronts lose the monotonicity and cause oscillations and periodic traveling wave solutions.In particular, the solution semiflows are not usually compact when the nonlocal term is incorporating into the equations. Therefore, it is not only more meaningful and valuable in theory and practice, but also more challengeable in mathematics to study such equations.This thesis is continuous to study such equations from the dynamical systems point of view, attempt to develop some new approaches for such equation and establish some new abstract results. For some specific models, investigate the influences of the time delay and spatial nonlocality, in particular, spatial nonlocality, on the dynamics of the equations and find some properties which were not reported previously. This thesis is mainly with traveling wave solutions and entire solutions in reaction-diffusion equations (systems) with nonlocal delays.

We firstly study the existence of traveling wave fronts of reaction-diffusion systems with nonlocal delays. By introducing different monotonicity conditions for the nonlinearity, and the concepts of G-compactness and M-continutity, we establish the existence of solutions for a class of abstract second-order mixed functional differential systems. Our methods are to use monotone iterations together with super- and subsolutions techniques and a non-standard ordering to develop a new monotone iteration scheme. We then apply the results to the corresponding wave systems and obtain the existence of traveling wave fronts. As a application for the main results, we carefully study the existence of traveling wave fronts for a single-species diffusive model with nonlocal delay and obtain some existence criteria of traveling wave fronts by choosing different kernels.

We further study the existence, uniqueness and asymptotic stability of traveling wave fronts of the quasi-monotone reaction advection diffusion equations with nonlocal delay.
We consider two case for the nonlinearity, that is, the monostable nonlinearity and the bistable nonlinearity. For the monostable case, the existence of traveling wave fronts
is obtained by using the results established for the existence of traveling wave fronts of reaction-diffusion systems with nonlocal delays and the asymptotic stability of traveling wave fronts with phase shift is proved via employing squeezing technique together with the comparison principle. Furthermore, we show a priori asymptotic behavior of traveling wave fronts in minus infinity and obtain the non-existence of traveling wave fronts at the same time, then the uniqueness of traveling wave fronts (up to translation) follows from the asymptotic stability. In particular, we find that the time delay can slow the spreading speed and the spatial nonlocality can increase the spreading speed. For the bistable case,we construct various pairs of super- and subsolutions and employ the comparison principle and the squeezing technique to prove that the equation has a unique traveling wave front (up to translation, the wave speed is also unique), which is monotonically increasing and globally asymptotically stable (with phase shift). The influence of advection on the propagation speed is also considered for both two cases. When applied to some population and epidemiological models, we obtained many meaningful results in practice.

Finally, using the increasing travelling wave solution estalished for the quasi-monotone reaction advection diffusion equations with nonlocal delay and the comparison argument,we prove the existence of entire solutions for the quasi-monotone reaction-diffusion equations with nonlocal delay. A key idea for the proof of the existence of entire solutions is to characterize the asymptotic behavior of the solutions as t → −∞ in term of appropriate subsolutions and supersolutions. For the bistable equation, we further show that such an entire solution is unique (up to space-time translations) and Lyapunov stable. For the monostable equation, by considering the mixing of traveling wave solutions and the solution independent of spatial variable, other new types of entire solutions are obtained. Moreover, some sufficient conditions are given to ensure the existence of the solutions independent of the spatial variable via using the theory of monotone dynamical systems. As applications of the main results, we investigate entire solutions two reaction-diffusion equations with nonlocal delay for the monostable nonlinearity and bistable nonlinearity,respectively.