Monotonicity of stable solutions in shadow systems

Ni, Wei-Ming; Poláčik, Peter; Yanagida, Eiji

doi:10.1090/S0002-9947-01-02880-X

Monotonicity of stable solutions in shadow systems
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by Wei-Ming Ni, Peter Poláčik and Eiji Yanagida PDF

Trans. Amer. Math. Soc. 353 (2001), 5057-5069 Request permission

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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