Monotonicity and Convergence Results in Order-preserving Systems in the Presence of Symmetry

This paper deals with various applications of two basic theorems in orderpreserving 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 di usion equations and systems, and show that stable traveling waves and pseudo-traveling waves have monotone pro les and, conversely, that monotone traveling waves and pseudotraveling waves are stable with asymptotic phase. We also discuss pseudo-traveling waves for equations of surface motion.