Journal of the European Mathematical Society

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Volume 18, Issue 1, 2016, pp. 195–223
DOI: 10.4171/JEMS/588

Published online: 2015-12-16

Group actions on monotone skew-product semiflows with applications

Feng Cao[1], Mats Gyllenberg[2] and Yi Wang[3]

(1) Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China
(2) University of Helsinki, Finland
(3) University of Scinece and Technology of China, Hefei, Anhui, China

We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group $G$-action has been considered on a strongly monotone skew-product semiflow. Here we relax the strong monotonicity and compactness requirements, and establish a theory concerning symmetry or monotonicity properties of uniformly stable 1-cover minimal sets. We then apply this theory to show rotational symmetry of certain stable entire solutions for a class of nonautonomous reaction-diffusion equations on $\mathbb R^n$, as well as monotonicity of stable traveling waves of some nonlinear diffusion equations in time-recurrent structures including almost periodicity and almost automorphy.

Keywords: Monotone skew-product semiflows, group actions, rotational symmetry, reaction-diffusion equations, traveling waves

Cao Feng, Gyllenberg Mats, Wang Yi: Group actions on monotone skew-product semiflows with applications. J. Eur. Math. Soc. 18 (2016), 195-223. doi: 10.4171/JEMS/588