Systems by the Method of Quasisolutions
Vatsala, A. S.
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Recently  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  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 , 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  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.