Reaction-Diffusion Inequalities in Cones
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Recently there has been a growing interest in the study of nonlinear reaction-diffusion equations [2,3,4,7] because of the fact examples of such equations occur in population genetics [2,5,12,13], nuclear and chemical reactors [2,7,8], conduction of nerve impulses [1,7,15], and several other biological models [1,6,15]. As is the case of ordinary differential equations [9,10], it is natural to expect that the theory of reaction-diffusion inequalities and comparison theorems will play a prominent role in this study. In this paper, we consider reaction-diffusion equations which are weakly coupled relative to an arbitrary cone. We prove a result on flow-invariance which is then utilized to obtain a useful comparison theorem and a theorem on differential inequalities. The results obtained are applied to simple reaction diffusion equations to derive positivity of solutions, upper and lower bounds and stability properties. Finally we demonstrate by means of a simple example that working with a suitable cone other than [see pdf for notation] is more advantageous in the investigation of equations of reaction-diffusion.