{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:31:49Z","timestamp":1772292709714,"version":"3.50.1"},"reference-count":0,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2011,3,1]],"date-time":"2011-03-01T00:00:00Z","timestamp":1298937600000},"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":[[2011,3]]},"abstract":"<jats:p> We consider a modified Cahn\u2013Hiliard equation where the velocity of the order parameter u depends on the past history of \u0394\u03bc, \u03bc being the chemical potential with an additional viscous term \u03b1u<jats:sub>t<\/jats:sub>, \u03b1\u22650. This type of equation has been proposed by P. Galenko et al. to model phase separation phenomena in special materials (e.g., glasses). In addition, the usual no-flux boundary condition for u is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The resulting boundary value problem is subject to suitable initial conditions and is reformulated in the so-called past history space. Existence of a variational solution is obtained. Then, in the case \u03b1&gt;0, we can also prove uniqueness and construct a strongly continuous semigroup acting on a suitable phase space. We show that the corresponding dynamical system has a (smooth) global attractor as well as an exponential attractor. In the case \u03b1=0, we only establish the existence of a trajectory attractor. <\/jats:p>","DOI":"10.3233\/asy-2010-1019","type":"journal-article","created":{"date-parts":[[2019,11,30]],"date-time":"2019-11-30T00:15:15Z","timestamp":1575072915000},"page":"123-162","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":8,"title":["Cahn\u2013Hilliard equations with memory and dynamic boundary conditions"],"prefix":"10.1177","volume":"71","author":[{"given":"Cecilia","family":"Cavaterra","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica \u201cF. Enriques\u201d, Universit\u00e0 degli Studi di Milano, 20133 Milano, Italy. E-mail: cecilia.cavaterra@unimi.it"}]},{"given":"Ciprian G.","family":"Gal","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: galc@missouri.edu"}]},{"given":"Maurizio","family":"Grasselli","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica \u201cF. Brioschi\u201d, Politecnico di Milano, 20133 Milano, Italy. E-mail: maurizio.grasselli@polimi.it"}]}],"member":"179","published-online":{"date-parts":[[2011,3,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2010-1019","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2010-1019","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,11]],"date-time":"2025-03-11T04:12:49Z","timestamp":1741666369000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2010-1019"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3]]},"references-count":0,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2011,3]]}},"alternative-id":["10.3233\/ASY-2010-1019"],"URL":"https:\/\/doi.org\/10.3233\/asy-2010-1019","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,3]]}}}