时标上动态方程的概周期问题研究

The development of almost-periodic problems is closely related to ordinary differential equations, stability theory, dynamical systems, functional differential equations, differential equations on Banach spaces, and a class of partial differential equations. Almost-periodic phenomenon has strong actual background, important scientific significance and wide applications. Up to now, research on almost-periodic phenomenon is mostly limited to continuous systems and little has been done about systems on time scales. The theory of time scales unifies the continuous and discrete analyses, combines the studies of differential equations and difference equations into one unified approach, and includes differential-difference equations and many others. Its development is of both theoretical importance and broad applications. We believe that we should further study almost-periodic problems on time scales...The proposed project will employ new theory for dynamic equations on measure chains to further study the basic theory of almost-periodic problems, establish some models of dynamic equations on time scales with practical background and study the qualitative behaviors of their almost-periodic type solutions and applications, and explore the relationship between almost-periodic problems on time scales and the set of real numbers. This project not only has innovation for pure theory, but also has great potential in applications.

概周期问题的发展密切地联系着常微分方程、稳定性理论、动力系统、泛函微分方程、Banach空间微分方程以及一类广泛的偏微分方程等，具有强烈的实际背景、重要的科学意义和广泛的应用；迄今的研究大多限于连续型系统，对时标系统很少研究。时标理论整合了连续与离散分析，它的研究不仅把微分方程和差分方程理论有机结合，而且也包含了兼有连续与离散现象共存的微分差分方程等，有理论上的重要性和广泛的应用。因此，我们有理由从时标的角度，继续深入研究概周期问题。..本项目将利用测度链上动态方程新理论，进一步研究概周期问题的基本理论；建立若干有实际背景的时标动态方程模型，研究其概周期类型解的定性性态及应用；探究一般时标与实数域上概周期问题之间的关系。此项目不仅有基础理论创新的价值，而且在技术应用有着巨大潜力。

概周期问题的发展密切地联系着常微分方程、稳定性理论、动力系统、泛函微分方程、Banach空间微分方程以及一类广泛的偏微分方程等，具有强烈的实际背景、重要的科学意义和广泛的应用。..四年来，我们课题组按研究计划开展工作，在可压缩向列型液晶方程在临界空间下的柯西问题、N(N>=3) 维可压缩Navier-Stokes-Possion系统在临界L^p型框架下全局解的大时间行为、半群是非紧的情况下具有非局部条件的Hilfer分数阶微分方程适度解的存在性、在全空间R^ N(N>=2)上的可压缩液晶流方程的解在L^p型临界空间中的最优衰减率、弱条件下的加权Stepanov伪概自守(周期)函数空间的完备性、时标上带有时滞和脉冲扰动项的N-种群非自治Lotka-Volterra竞争系统的持久性和概周期解、全空间R^ N(N>=2)上的可压缩液晶流方程在L^p型临界Besov空间上的不可压缩极限问题、一类分数阶积分微分中立型方程r类伪S渐近w-周期弱解的存在唯一性、一类时标上具有混合延迟的分流抑制细胞神经网络的加权伪概周期解的存在唯一性和全局指数稳定性、在临界泛函框架下变密度三维不可压缩磁流体力学（MHD）方程解的整体适定性、加权S^p-伪S-渐近周期函数所成空间的性质及其应用等方面开展了一系列创新性的研究工作。