Monotonicity and convergence results in  order-preserving systems in  the presence of symmetry

Toshiko Ogiwara; Hiroshi Matano

doi:10.3934/dcds.1999.5.1

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1999, Volume 5, Issue 1: 1-34. Doi: 10.3934/dcds.1999.5.1

This issue Previous Article Next Article Anti-periodic solutions to a class of non-monotone evolution equations

Monotonicity and convergence results in order-preserving systems in the presence of symmetry

Toshiko Ogiwara^1, and
Hiroshi Matano^2,

1.
Laboratory of Nonlinear Studies and Computation, Research Institute for Electronic Science, Hokkaido University, Kita-ku, Sapporo 060
2.
Graduate School of Mathematical Sciences, University of Tokyo, Komaba Tokyo, 153-8914

Revised: August 31, 1998

Published: December 1998

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Abstract

This paper deals with various applications of two basic theorems in order- preserving systems under a group action -- monotonicity theorem and convergence theorem. Among other things we show symmetry properties of stable solutions of semilinear elliptic equations and systems. Next we apply our theory to traveling waves and pseudo-traveling waves for a certain class of quasilinear diffusion equa- tions and systems, and show that stable traveling waves and pseudo-traveling waves have monotone profiles and, conversely, that monotone traveling waves and pseudo- traveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.

Keywords:

Order-preserving systems,
semilinear elliptic equations,
traveling waves and pseudo-traveling waves,
quasilinear diffusion equations.

Mathematics Subject Classification: 35K40, 35K57.

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