Prevalent Behavior of Strongly Order Preserving Semiflows |
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Authors: | Germán A Enciso Morris W Hirsch Hal L Smith |
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Institution: | (1) Mathematical Biosciences Institute, Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA;(2) Department of Mathematics, University of California, Berkeley, CA 94720, USA;(3) Department of Mathematics, Arizona State University, Tempe, AZ 85287, USA |
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Abstract: | Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward
an equilibrium or toward the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results
in terms of the measure-theoretic notion of prevalence, developed in Christensen (Israel J. Math., 13, 255–260, 1972) and Hunt et al. (Bull. Am. Math. Soc., 27, 217–238, 1992). For monotone reaction–diffusion systems with Neumann boundary conditions on convex domains, we show the
prevalence of the set of continuous initial conditions corresponding to solutions that converge to a spatially homogeneous
equilibrium. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is
given to the measurability of the various sets involved. |
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Keywords: | Strong monotonicity prevalence quasi-convergence reaction– diffusion measurability |
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