抽象时滞发展方程周期解的存在性及渐近性态

项目名称： 抽象时滞发展方程周期解的存在性及渐近性态

项目编号： No.11261053

项目类型： 地区科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 李永祥

作者单位： 西北师范大学

项目金额： 50万元

中文摘要： 时滞偏微分方程周期解问题是非线性分析与偏微分方程中人们非常关注的问题，抽象空间发展方程是含时间变量的偏微分方程的概括描述，抽象时滞发展方程周期解问题的研究有望使一些具体的时滞偏微分方程周期解问题获得一般的解决，具有重要的理论与应用意义。在本项目中，我们拟用算子半群理论与非线性分析的理论工具与方法深入地研究抽象时滞发展方程周期解问题，就各种应用情形分别获得周期解的存在性与多重性、唯一性与渐进稳定性的等结果，并将获得的抽象结果应用于一些具体的时滞偏微分方程，检验我们的结果的适用性。特别应用我们的抽象结果进一步研究时滞抛物型偏微分方程与时滞电报方程正周期解的存在性与渐近稳定性。

中文关键词： 发展方程；时滞；周期解；线性算子半群；渐近稳定性

英文摘要： The periodic problem of partial differential equation with delays is an important research topic in the field of nonlinear analysis and partial differential equations and it has attracted many researchers' attention and concern. Many types of partial differential equations with time argument can be written summarily to the form of abstract evolution equations. The study of the problem on the abstract evolution equations with delays is hopeful to obtain the general solution for the periodic problems of some concrete partial differential equations with delays. Hence, it has important significance in theory and applications. In this project, we will research the periodic problem of abstract evolution equations with delays by the theory of operators semigroups and the methods of nonlinear analysis. For various types of abstract evolution equations with delays proposed from applications, we will try to obtain the existence, multiplicity, uniqueness and asymptotic stability results of periodic solutions. We will apply our abstract theorems to some concrete partial differential equations with delays to examine the applicability of the abstract theorems. Especially by applying our abstract theorems, we will further discuss the existence and asymptotic stability of positive periodic solutions of parabolic delay equations

英文关键词： evolution equation；delay；periodic solution；semigroup of linear operators；asymptotic stability