ABSTRACT
Chapter 2 is devoted to the study of second-order parabolic initial boundary value problems (IBVPs) relative to the extension of the monotone iterative technique and the method of generalized quasilinearization to such parabolic problems. Section 2.2 states the IB VP and provides the required compari son results. In Section 2.3, we go directly to develop the monotone iterative technique in the general set-up and point out the various special cases that can be derived from the general results considered. The method of gener alized quasilinearization forms the content of Section 2.4 together with the details of interesting particular cases. Section 2.5 is dedicated to weakly cou pled mixed monotone parabolic systems and corresponding elliptic systems of BVPs. Utilizing the theory of monotone flows and the ideas of the reflec tion operator, the convergence of solutions of mixed monotone systems to the solutions of the corresponding elliptic systems is investigated. In Section 2 .6 , we consider the comparison results of weakly coupled parabolic systems of IBVPs in a very general framework so as to bring out ideas involved in such a set-up. The method of vector Lyapunov functions is developed for weakly coupled parabolic systems of IBVPs in Section 2 .7 , which leads to stability theory of solutions of such systems. An example is worked out in detail, which demonstrates how the diffusion, convection and reaction terms contribute to stability in various ways and play an important role.
