Coupled Nonlinear Parabolic Systems with Time Delays

A coupled system of parabolic equations with time delays is investigated by the method of upper-lower solutions. It is shown by using upper and lower solutions as coupled initial iterations that the corresponding sequences converge monotonically to a unique solution of the system. This convergence leads to an existence-comparison theorem which is useful for studying the asymptotic behavior of the solution. This method is extended to a coupled system of integro-differential equations and systems of parabolic-ordinary equations. Applications are given to some model problems in ecology and nuclear engineering, including the stability property of a steady-state solution.