Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations

We consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. We then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in with nonlinearities depending on the gradient of the solution. We consider as well systems of reaction–diffusion equations in and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in . We further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn–Hilliard equation.