{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:30:40Z","timestamp":1777447840074,"version":"3.51.4"},"reference-count":45,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2017,1,26]],"date-time":"2017-01-26T00:00:00Z","timestamp":1485388800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2017,1,26]]},"abstract":"<jats:p>Recently, in Bonfoh and Enyi [ Commun. Pure Appl. Anal. 15 2016 , 1077\u20131105], we considered the conserved phase-field system [Formula: see text] in a bounded domain of [Formula: see text], [Formula: see text], where [Formula: see text] is a relaxation time, [Formula: see text] is the viscosity parameter, [Formula: see text] is the heat capacity, \u03d5 is the order parameter, u is the absolute temperature and [Formula: see text] 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 [Formula: see text]. 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 [Formula: see text]. To establish the existence of inertial manifolds of dimension independent of the two parameters \u03b4 and \u03b5, we require \u03b5 to be dominated from above by \u03b4. This work shows the convergence of the dynamics of the above mentioned problem to the one of the Cahn\u2013Hilliard equation, improving and extending some previous results.<\/jats:p>","DOI":"10.3233\/asy-161395","type":"journal-article","created":{"date-parts":[[2017,1,27]],"date-time":"2017-01-27T11:14:40Z","timestamp":1485515680000},"page":"97-148","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":2,"title":["The Cahn\u2013Hilliard equation as limit of a\u00a0conserved phase-field system"],"prefix":"10.1177","volume":"101","author":[{"given":"Ahmed","family":"Bonfoh","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O.\u00a0Box\u00a0546, Dhahran 31261, Saudi Arabia. E-mails:\u00a0,\u00a0"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cyril D.","family":"Enyi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, P.O.\u00a0Box\u00a0546, Dhahran 31261, Saudi Arabia. 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