Abstract
Recently, in Bonfoh and Enyi [Commun. Pure Appl. Anal. 15 2016, 1077-1105], we considered the conserved phase-field system
{tau phi t - Delta(delta phi(t) - Delta phi + g(phi) - u) = 0.
epsilon u(t) + phi(t) - Delta u = 0.
in a bounded domain of R-d, d = 1, 2, 3, where tau > 0 is a relaxation time, delta > 0 is the viscosity parameter, epsilon is an element of (0, 1] is the heat capacity, phi is the order parameter, u is the absolute temperature and g : R -> R is a nonlinear function. The system is subject to the boundary conditions of either periodic or Neumann type. We proved a well-posedness result, the existence and continuity of the global and exponential attractors at epsilon = 0. Then, we proved the existence of inertial manifolds in one space dimension, and in the case of two space dimensions in rectangular domains. Stability properties of the intersection of inertial manifolds with a bounded absorbing set were also proven. In the present paper, we show the above-mentioned existence and continuity properties at (epsilon, delta) = (0, 0). To establish the existence of inertial manifolds of dimension independent of the two parameters delta and epsilon, we require epsilon to be dominated from above by delta. This work shows the convergence of the dynamics of the above mentioned problem to the one of the Cahn-Hilliard equation, improving and extending some previous results.