Monotonicity of stable solutions in shadow systems
Authors:
WeiMing Ni, Peter Polácik and Eiji Yanagida
Journal:
Trans. Amer. Math. Soc. 353 (2001), 50575069
MSC (2000):
Primary 35K50; Secondary 35B35
Published electronically:
July 25, 2001
MathSciNet review:
1852094
Fulltext PDF Free Access
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Abstract: A shadow system appears as a limit of a reactiondiffusion system in which some components have infinite diffusivity. We investigate the spatial structure of its stable solutions. It is known that, unlike scalar reactiondiffusion 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 nonmonotone 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 .
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Additional Information
WeiMing 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 9808578, Japan
Email:
yanagida@math.tohoku.ac.jp
DOI:
http://dx.doi.org/10.1090/S000299470102880X
PII:
S 00029947(01)02880X
Received by editor(s):
January 27, 2000
Published electronically:
July 25, 2001
Article copyright:
© Copyright 2001
American Mathematical Society
