Large time behavior of  a conserved phase-field system

Ahmed  Bonfoh; Cyril D.  Enyi

doi:10.3934/cpaa.2016.15.1077

Article Contents

2016, Volume 15, Issue 4: 1077-1105. Doi: 10.3934/cpaa.2016.15.1077

This issue Previous Article Next Article Nonlinear noncoercive Neumann problems

Large time behavior of a conserved phase-field system

Ahmed Bonfoh^1, and
Cyril D. Enyi^1,

1.
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O. Box 546, Dhahran 31261

Received: December 2015

Revised: February 2016

Published: July 2016

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Abstract

We investigate the large time behavior of a conserved phase-field system that describes the phase separation in a material with viscosity effects. We prove a well-posedness result, the existence of the global attractor and its upper semicontinuity, when the heat capacity tends to zero. Then we prove the existence of inertial manifolds in one space dimension, and for the case of a rectangular domain in two space dimension. We also construct robust families of exponential attractors that converge in the sense of upper and lower semicontinuity to those of the viscous Cahn-Hilliard equation. Continuity properties of the intersection of the inertial manifolds with bounded absorbing sets are also proven. This work extends and improves some recent results proven by A. Bonfoh for both the conserved and non-conserved phase-field systems.

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Mathematics Subject Classification: Primary: 35B25, 35B41, 37L25; Secondary: 82C26.

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