Existence and stability of periodic solutions for parabolic systems with time delays

This paper is concerned with the existence and stability time-periodic solutions for a class of coupled parabolic equations with time delay, and time delays may appear in the nonlinear reaction functions. The existence of time-periodic solutions is for a class of locally Lipschitz continuous reaction functions without any quasimonotone requirement. Our approach to the problem is by the method of upper and lower solution and using Schauder fixed point theorem. Some methods for proving the stability of the periodic solution are also given. The results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system. Finally, a model arising from chemistry is used to illustrate the obtained results.