Existence And Uniqueness Of Periodic Solutions For Partial Differential Equations With Delay

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.