Dynamics of Nonlinear Parabolic Systems with Time Delays

The dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions. It is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system. Under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system. The same conclusions are obtained for a coupled system of parabolic-ordinary equations with time delays. Applications are given to three model problems arising from ecology and nuclear engineering. These model problems possess multiple steady-state solutions and sufficient conditions are given to ensure the stability and instability of these solutions.