Existence of coupled quasi-solutions of systems of nonlinear reaction-diffusion equations

Abstract

Systems of nonlinear parabolic initial boundary value problems arise in many applications, such as epidemies, ecology, biochemistry, biology, chemical and nuclear engineering. Constructive methods of proving existence results for such problems, which can also provide numerical procedures for the computation of solutions, are of greater value than theoretical existence results. The method of upper and lower solutions coupled with monotone iterative technique has been employed successfully to prove existence of multiple solutions of nonlinear reaction-diffusion equations, in special cases, by various authors [335, 10, 11, 15, 181. Recently, in [6, 171 weakly coupled systems of reaction-diffusion equations, when the nonlinear terms are independent of gradient terms, are discussed and some special type of results are obtained. We, in this paper, investigate general systems of nonlinear reaction-diffusion problems when the nonlinear terms possess a mixed quasi-monotone property. We discuss a very general situation and obtain coupled extremal quasi-solutions, which in special cases reduce to minimal and maximal solutions. We shall also indicate how one-step cyclic monotone iterative schemes can be generated which yield accelerated rate of convergence of iterates. This work is in the spirit of our recent paper [12] for elliptic systems.