【作者】 王长有；

【导师】 李树勇；

【作者基本信息】 四川师范大学 ， 运筹学与控制论， 2004， 硕士

【摘要】 本文研究几类含时滞的偏微分方程周期解问题。在现实世界中许多现象都是与过去有联系的，用时滞偏微分方程来刻划显得更真实，更接近实际。含时滞的偏微分方程的解的表现比不含时滞的偏微分方程的解的表现要复杂得多。时滞可影响方程解的存在性、稳定性，可导致周期、拟周期等更为复杂的现象，因而在研究上更加困难。本文中，我们主要研究几类含时滞的偏微分方程周期解的存在唯一性。 首先，我们研究含时滞的周期抛物型方程组应用上、下解方法及相应的单调迭代方法，证明：如果反应项混拟单调且方程存在周期上、下解，则方程一定存在周期拟解，并且拟解构成的区间是一个吸引子。对一些具体的方程，在某些条件下，周期拟解恰好是方程的周期解，并以一个生物学模型说明了取得结果的意义。 其次，我们考虑如下时滞反应扩散方程通过构造上，下控制函数，就反应项非单调情形进行探索，建立了周期解存在唯一性的充分条件，推广了己有的结果.最后，我们进一步研究时滞反应扩散系统的边值问题利用不动点定理，上、下解方法及算子理论，得到方程周期解的存在性.更多还原

【Abstract】 This paper studies partial differential equations with delay. Because many of processes in realistic world have connections with the past, it is more realistic to model a real system by partial differential equations with delay. The behaviour of solutions of the partial differential equations with delays is a lot more complex than that without delay. The delay can affect the existence and stability of solutions as well as leads to the periodic, quasi-periodic and more complex phenomena. So it is more difficult to study the partial differential equations with delay. In this paper we mainly study the existence and uniqueness of the periodic solutions for several classes of the special partial differential equations with delay.Firstly, we study the periodic parabolic equations with delaysBy using the method of upper and lower solutions and its associated monotone iterations, it is shown that the reaction function in the system possesses a quasi-monotone property and the periodic boundary value system has a pair of coupled T-upper and lower solutions then the periodic boundary value problem has a pair of periodic quasi-solutions and the sector between the quasi-solutions is an attractor of the delayed periodic parabolic system. Under some additional conditions the periodic quasi-solutions are exactly true periodic solutions of the system. Finally, a model arising from ecology is given to illustrate the obtained results.Secondly, we study the reaction-diffusion systems with delay.By constructing the upper and lower control function, it is shown that periodic solutions of this system exist and are unique when reaction-term is not monotone. And some known results are extended.Finally, we further consider the following boundary valuable problem of the reaction-diffusion system with time delay.The existence of periodic solutions for this system by the periodic upper and lower solution method and fixed point theory are investigated.更多还原