Monotone iterative methods and Schwarz methods for nonlinear parabolic PDE with time delay

Its well known that the Schwarz alternating method proved to be feasible and powerful approach to solve elliptic or parabolic PDEs over multi overlapped subdomains. Recently Schwarz method proved to be very effective method when embedded within a well established solution methods such as monotone iterative method( or the method of lower and upper solution method). In this work we present the proofs of convergence of additive and multiplicative Schwarz alternating method for non linear parabolic equation where the reaction function involves a time delay function.