一类时滞反应扩散系统行波解的稳定性

The asymptotic behavior in long time of solutions of the corresponding Cauchy problems is determined by traveling wave solutions generally, which is a kind of steady-state solutions of reaction-diffusion equations. It has an extensive application background in Ecology, Epidemiology and much fields. By using the theory of (partial) functional differential equations, we are concerned with the stability of mono-bistable traveling waves for a class of reaction-diffusion systems with (nonlocal)delay in this project. The comparison principle does not hold because of the scarcity of quasi-monotonicity in systems; the spectral analysis are no effect for the stability of multidimensional case when the spectrum gap disappears due to the effect of the transverse diffusion. Moreover, dynamic behavior of equations may be led to change because of the time-delay and appearance of coupling; the normally weighted energy estimate can not be applied to monostable traveling waves in the case of critical speed because there is no good enough regularity for the solutions caused by spatial non-locality and therefore it is not suitable any longer by using the frequent methods and normal theory to solve the stability of traveling wave solutions for classical reaction-diffusion equations. Based on the above fact, we are expected to establish some innovative abstract results and apply to some epidemic models through the development of new research methods, which will be used to explain and control the propagation of infection disease. Therefore, it is significant on theory and practice to study the stability of traveling waves of reaction-diffusion systems with (nonlocal)delay and with or without quasi-monotonicity.

行波解作为反应扩散方程的一种稳态解, 通常决定相应 Cauchy问题解的长时间渐近行为，在生态学、传染病学等领域有着广泛的应用背景。本项目借助(偏)泛函微分方程等理论研究一类(非局部)时滞反应扩散系统单(双)稳行波解的稳定性。由于系统拟单调性缺失时比较原理不成立；高维情形下由于横截扩散的影响，谱缝隙消失使得谱分析失效。另外，时滞和系统耦合的出现可能引起方程动力学行为的变化；空间非局部项使解的正则性不够好,通常的加权能量估计不能直接运用到临界波速下的单稳波，从而一些研究经典反应扩散方程行波解稳定性的标准理论和常用方法不再适用。本项目可望通过发展新方法，建立一些创新性的抽象结果并运用到具体的传染病模型，对解释和控制传染病的传播等实际问题提供理论依据。因此，对(非)拟单调(非局部)时滞系统行波解稳定性的研究，具有重要的理论意义和应用价值。

反应扩散方程的行波理论中, 作为主要问题之一的行波解的稳定性一直是重点和难点，在传染病学、人口动力学等领域有重要的理论意义和应用价值。本项目借助半线性抛物方程的几何理论、算子半群理论、非线性泛函分析、(偏)泛函微分方程等理论研究了一类时滞反应扩散标量方程和系统行波解的稳定性、唯一性和渐近行为等。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)建立了一类拟单调时滞系统双稳行波解的单调性、唯一性和李雅普诺夫稳定性。(2)研究了一类时空非局部时滞的人口动力学模型单稳行波解的全局指数稳定性。(3)建立了一类带对流项和非局部时滞的人口动力学模型单稳波前解的全局指数稳定性。(4)分别研究了一类时滞的SIR模型和具有分布时滞的非拟单调传染病系统非单调行波解的指数稳定性和唯一性。(5)研究了一类拟单调时滞传染病系统临界波速下单稳波前解的全局代数稳定性。