ABSTRACT
This chapter looks at the asymptotic behaviour of solutions to the phase-field equations and coupled Cahn-Hilliard equations as t → ∞. It considers an unusual type of nonlinear elliptic boundary value problem with nonlocal terms and constraints arising from the study of stationary problems of the coupled Cahn-Hilliard equations and also of the phase-field equations. A new technique is introduced to deal with problems in one space dimension. The problems are reduced to finding intersection points of two analytic functions of one variable. The chapter combines the results earlier obtained to conclude the convergence of solution of the corresponding evolution equations to an equilibrium as t → ∞ in one space dimension. The chapter also looks at the study of infinite-dimensional dynamical systems associated with the phase-field equations.
