时滞非局部扩散方程的单稳行波解及其稳定性

By the effect of delay and spatial non-locality, the changes of dynamic behavior of equations and the difficulties in theoretical investigation are both led to appear. For example, the wave speed is influnenced by delay and nonlocal dispersal, and it is not suitable any longer by using the frequent methods and normal theory for solving the problems of traveling wave solutions in classical reaction-diffusion equations because the nonlocal convolution operator has poor smoothness. Based on the above facts, we are concerned with the monostable traveling waves and their stability of scalar equations and systems with delay and nonlocal dispersal in this project. Especially, it is more difficult to investigate the stability of critical waves of nonlocal dispersal systems with delay and without quasi-monotonicity. The usually weighted energy method can not be directly applied to the critical waves for the appearance of nonlocal diffusion term and the loss of quasi-monotonicity in systems; and it is not easy to get the decay estimates of elementary solutions of ordinary functional differential equations because of the coupling of different unknown functions. Therefore, it brings a new challenge to develop and improve theory of traveling waves of reaction-diffusion equations with delay. Furthermore, stimulated by the propagation theory in classical reaction-diffusion equations, it is significant on theory and practice to go on concerning in the impact on traveling waves derived from delay and nonlocal dispersal.

时滞和空间非局部作用不仅引起方程动力学行为的变化，而且带来理论研究上的困难，例如时滞和非局部扩散影响波的传播速度；非局部卷积算子光滑性低、使研究经典反应扩散方程行波解的一些标准理论和常用方法不再适用。基于此，本项目研究时滞非局部扩散标量方程和系统的单稳行波解及其稳定性。其中，非拟单调的时滞非局部扩散系统临界波速下单稳行波解的稳定性研究更加困难，因为非局部扩散项的出现和系统拟单调性的缺失，使通常的加权能量方法不再适用于临界波速的情形；不同未知函数的耦合使常泛函微分方程基本解的衰减估计不易得到，这为发展和完善时滞反应扩散方程的行波解理论提出了新的挑战。另外，受经典反应扩散方程中得出的传播理论的启发，继续研究时滞和非局部扩散对行波解产生的影响，具有重要的理论意义和实践价值。

由于时滞和空间非局部作用会引起方程动力学行为的变化，受经典反应扩散方程传播理论(扩散项和反应项都对波的传播产生影响)的启发，通过对具体传染病模型和种群动力学模型行波解的研究建立适用于一般时滞非局部扩散标量方程和系统的行波理论，具有重要的理论意义和实践价值。特别是将一些离散时滞的非局部扩散标量方程的行波解研究结果推广到非局部扩散的非局部时滞系统，丰富并完善了非局部扩散方程行波解的研究工作。. 本项目借助非线性泛函分析、常(偏)泛函微分方程理论和算子半群理论等研究了几类时滞非局部扩散方程行波解的存在性、唯一性、渐近行为和稳定性等。目前已完成主要研究内容，并取得了一些研究成果，达到了项目的预期目标。主要研究内容和成果包括：(1)建立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（不）存在性、有界性和渐近行为。(2)研究了一类具有静止期的时滞种群动力学模型单稳行波解的稳定性和一类非拟单调的空间非局部时滞标量方程单稳行波解的稳定性。(3)建立了一类带输入项和非局部扩散项的非局部时滞 SIR 模型行波解的（不）存在性和渐近行为。(4)研究了一类带复发性和非局部扩散的时空时滞传染病系统行波解的（不）存在性和渐近行为。(5) 研究了一类具有空间非局部效应的（非）拟单调时滞非局部扩散系统非临界波速下单稳行波解的稳定性。(6) 研究了一类带输入项和非局部扩散项的非局部时滞 SIR 模型(非)临界波速下单稳行波解的稳定性。