Systems by the Method of Quasisolutions

Abstract

**Please note that the full text is embargoed** ABSTRACT: Recently [10] the method of lower and upper solutions has been extended to systems of reaction diffusion equations which has become very useful in dealing with applications. This extension depends crucially on a certain property known as quasimonotone nondecreasing property [8] without which the results fail under natural definition of lower and upper solutions. When the quasimonotone property does not hold but a certain mixed quasimonotone property is satisfied, which is the case in several applications [7], the method of quasisolutions is more suitable [2,4,6,9]. All these results utilize monotone iterative technique. When no monotone condition holds one can also get just existence results [5] assuming Müller's type of lower and upper solutions. However in this case monotone technique fails. In this paper, we discuss the asymptotic stability of the stationary solution of reaction-diffusion systems. We employ the method of quasisolutions and monotone technique.