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Monotonicity of stable solutions in shadow systems

Authors: Wei-Ming Ni, Peter Polácik and Eiji Yanagida
Journal: Trans. Amer. Math. Soc. 353 (2001), 5057-5069
MSC (2000): Primary 35K50; Secondary 35B35
Published electronically: July 25, 2001
MathSciNet review: 1852094
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Abstract: A shadow system appears as a limit of a reaction-diffusion system in which some components have infinite diffusivity. We investigate the spatial structure of its stable solutions. It is known that, unlike scalar reaction-diffusion equations, some shadow systems may have stable nonconstant (monotone) solutions. On the other hand, it is also known that in autonomous shadow systems any nonconstant non-monotone stationary solution is necessarily unstable. In this paper, it is shown in a general setting that any stable bounded (not necessarily stationary) solution is asymptotically homogeneous or eventually monotone in $x$.

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Additional Information

Wei-Ming Ni
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Peter Polácik
Affiliation: Institute of Applied Mathematics, Comenius University, Mlynská dolina, 842 15 Bratislava, Slovakia

Eiji Yanagida
Affiliation: Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Received by editor(s): January 27, 2000
Published electronically: July 25, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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