We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group -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 , as well as monotonicity of stable traveling waves of some nonlinear diffusion equations in time-recurrent structures including almost periodicity and almost automorphy.
Cite this article
Feng Cao, Mats Gyllenberg, Yi Wang, Group actions on monotone skew-product semiflows with applications. J. Eur. Math. Soc. 18 (2016), no. 1, pp. 195–223DOI 10.4171/JEMS/588