Group actions on strongly monotone dynamical systems
It is well known that second order parabolic partial differential equations satisfying the strong comparison principle generate strongly monotone flows on appropriate function spaces. For such systems, almost all points whose trajectories are relatively compact are quasiconvergent, that is, their \(\omega\)-limit sets contain only equilibria. The purpose of the paper is to investigate strongly monotone flows on which a compact connected group acts monotonely and equivariantly. The main results state, loosely speaking, that almost all trajectories eventually symmetrize. More precisely, assume for simplicity that all points \(x\in X\) have relatively compact trajectories. Then the following holds: (1) every stable equilibrium is symmetric; (2) if X is a separable Banach space then the set Y of points whose \(\omega\)-limit sets contain only symmetric equilibria is residual in X; (3) if the flow is order- compact then Y is open and dense in X.
second order parabolic partial differential equations
scientific article; zbMATH DE number 4046751
scientific article