Systems of Parabolic Equations with Continuous and Discrete Delays

In this paper we investigate the global existence and the dynamics of a coupled system of nonlinear parabolic equations where the nonlinear ‘‘reaction function’’ may depend on both continuous infinite or finite.and discrete delays. It is shown that if the reaction function is locally Lipschitz continuous and the system possesses a pair of coupled upper and lower solutions then there exists a unique global solution to the system without any quasimonotone condition on the reaction function. For systems with mixed quasimonotone reaction functions we use the monotone method to establish more dynamic property of the parabolic system in terms of the quasisolutions of the corresponding elliptic system. This approach yields a global.attractor of the parabolic system, and under some additional conditions this attractor leads to the existence and asymptotic stability of a solution of the elliptic system. Application to three model problems in population dynamics and chemical reactions is given.