Group Actions on Monotone Skew-Product Semiflows with Applications

TL;DR

This paper develops a generalized theory for monotone skew-product semiflows with group actions, relaxing previous constraints, and applies it to demonstrate symmetry and monotonicity properties of solutions in reaction-diffusion and diffusion equations.

Contribution

It extends existing frameworks by relaxing assumptions on monotonicity and group compactness, providing new insights into symmetry and stability of solutions.

Findings

Rotational symmetry of certain stable entire solutions

Monotonicity of stable travelling waves in time recurrent structures

Application to non-autonomous reaction-diffusion equations

Abstract

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 requirement of strong monotonicity of the skew-product semiflows and the compactness of $G$, 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 non-autonomous reaction-diffusion equations on $\RR^n$, as well as monotonicity of stable travelling waves of some nonlinear diffusion equations in time recurrent structures including almost periodicity and almost automorphy.