一类时滞反应扩散系统行波解的稳定性

项目名称： 一类时滞反应扩散系统行波解的稳定性

项目编号： No.11301241

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 杨赟瑞

作者单位： 兰州交通大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 时滞；行波；稳定性；；

英文摘要： 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 esta

英文关键词： Delay；traveling waves；stability；；