Existence and Asymptotic Behavior of Reaction-Diffusion Systems Via Coupled Quasi-Solutions

Abstract

**Please note that the full text is embargoed** ABSTRACT: In the study of comparison theorems, existence of extremal solutions and monotone iterative techniques for differential systems a property called quasimonotone property is necessary [2,7,10]. However, there are several physical situations wherein such a property is not satisfied [1]. This difficulty has been overcome by introducing the notion of quasi-solutions [3,8,9]. In this paper we consider the reaction-diffusion system in which quasi-monotone property is not satisfied but a mixed quasimonotone property holds. By utilizing fruitfully the notion of quasi-solutions we prove the existence of coupled maximal and minimal solutions. For this purpose we exploit the monotone iterative technique. We then offer methods of constructing explicit coupled upper and lower solutions from which we deduce the asymptotic behaviour of solutions. Our results are in the spirit of similar results given in [1,5,10] and shed much light on the whole situation.