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 .

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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

Email:
yanagida@math.tohoku.ac.jp

DOI:
https://doi.org/10.1090/S0002-9947-01-02880-X

Received by editor(s):
January 27, 2000

Published electronically:
July 25, 2001

Article copyright:
© Copyright 2001
American Mathematical Society