{"id":"https://openalex.org/W233901638","doi":"https://doi.org/10.3233/asy-2010-1019","title":"Cahn\u2013Hilliard equations with memory and dynamic boundary conditions","display_name":"Cahn\u2013Hilliard equations with memory and dynamic boundary conditions","publication_year":2011,"publication_date":"2011-03-01","ids":{"openalex":"https://openalex.org/W233901638","doi":"https://doi.org/10.3233/asy-2010-1019","mag":"233901638"},"language":"en","primary_location":{"id":"doi:10.3233/asy-2010-1019","is_oa":false,"landing_page_url":"https://doi.org/10.3233/asy-2010-1019","pdf_url":null,"source":{"id":"https://openalex.org/S141283892","display_name":"Asymptotic Analysis","issn_l":"0921-7134","issn":["0921-7134","1875-8576"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318577","host_organization_name":"IOS Press","host_organization_lineage":["https://openalex.org/P4310318577"],"host_organization_lineage_names":["IOS Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Asymptotic Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://hdl.handle.net/2434/167367","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5078312828","display_name":"Cecilia Cavaterra","orcid":"https://orcid.org/0000-0002-2754-7714"},"institutions":[{"id":"https://openalex.org/I189158943","display_name":"University of Milan","ror":"https://ror.org/00wjc7c48","country_code":"IT","type":"education","lineage":["https://openalex.org/I189158943"]}],"countries":["IT"],"is_corresponding":true,"raw_author_name":"Cecilia Cavaterra","raw_affiliation_strings":["Dipartimento di Matematica \u201cF. Enriques\u201d, Universit\u00e0 degli Studi di Milano, 20133 Milano, Italy. E-mail: cecilia.cavaterra@unimi.it"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica \u201cF. Enriques\u201d, Universit\u00e0 degli Studi di Milano, 20133 Milano, Italy. E-mail: cecilia.cavaterra@unimi.it","institution_ids":["https://openalex.org/I189158943"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004351524","display_name":"Ciprian G. Gal","orcid":"https://orcid.org/0000-0002-6408-2565"},"institutions":[{"id":"https://openalex.org/I76835614","display_name":"University of Missouri","ror":"https://ror.org/02ymw8z06","country_code":"US","type":"education","lineage":["https://openalex.org/I76835614"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ciprian G. Gal","raw_affiliation_strings":["Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: galc@missouri.edu"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Missouri, Columbia, MO 65211, USA. E-mail: galc@missouri.edu","institution_ids":["https://openalex.org/I76835614"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5054462385","display_name":"Maurizio Grasselli","orcid":"https://orcid.org/0000-0003-2521-2926"},"institutions":[{"id":"https://openalex.org/I93860229","display_name":"Politecnico di Milano","ror":"https://ror.org/01nffqt88","country_code":"IT","type":"education","lineage":["https://openalex.org/I93860229"]}],"countries":["IT"],"is_corresponding":false,"raw_author_name":"Maurizio Grasselli","raw_affiliation_strings":["Dipartimento di Matematica \u201cF. Brioschi\u201d, Politecnico di Milano, 20133 Milano, Italy. E-mail: maurizio.grasselli@polimi.it"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dipartimento di Matematica \u201cF. Brioschi\u201d, Politecnico di Milano, 20133 Milano, Italy. E-mail: maurizio.grasselli@polimi.it","institution_ids":["https://openalex.org/I93860229"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5078312828"],"corresponding_institution_ids":["https://openalex.org/I189158943"],"apc_list":null,"apc_paid":null,"fwci":0.4812,"has_fulltext":false,"cited_by_count":24,"citation_normalized_percentile":{"value":0.61221712,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"71","issue":"3","first_page":"123","last_page":"162"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9986000061035156,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9900000095367432,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/cahn\u2013hilliard-equation","display_name":"Cahn\u2013Hilliard equation","score":0.5864666104316711},{"id":"https://openalex.org/keywords/dynamic-equation","display_name":"Dynamic equation","score":0.45523715019226074},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.44395071268081665},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37094634771347046},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.35071712732315063},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3401537537574768},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.27187293767929077},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.11865180730819702},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.09311184287071228}],"concepts":[{"id":"https://openalex.org/C24822716","wikidata":"https://www.wikidata.org/wiki/Q5017440","display_name":"Cahn\u2013Hilliard equation","level":3,"score":0.5864666104316711},{"id":"https://openalex.org/C2989370511","wikidata":"https://www.wikidata.org/wiki/Q18381590","display_name":"Dynamic equation","level":3,"score":0.45523715019226074},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.44395071268081665},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37094634771347046},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.35071712732315063},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3401537537574768},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.27187293767929077},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.11865180730819702},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.09311184287071228},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3233/asy-2010-1019","is_oa":false,"landing_page_url":"https://doi.org/10.3233/asy-2010-1019","pdf_url":null,"source":{"id":"https://openalex.org/S141283892","display_name":"Asymptotic Analysis","issn_l":"0921-7134","issn":["0921-7134","1875-8576"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310318577","host_organization_name":"IOS Press","host_organization_lineage":["https://openalex.org/P4310318577"],"host_organization_lineage_names":["IOS Press"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Asymptotic Analysis","raw_type":"journal-article"},{"id":"pmh:oai:air.unimi.it:2434/167367","is_oa":true,"landing_page_url":"http://hdl.handle.net/2434/167367","pdf_url":null,"source":{"id":"https://openalex.org/S4306400516","display_name":"Archivio Istituzionale della Ricerca (Universita Degli Studi Di Milano)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I189158943","host_organization_name":"University of Milan","host_organization_lineage":["https://openalex.org/I189158943"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:re.public.polimi.it:11311/609312","is_oa":false,"landing_page_url":"http://hdl.handle.net/11311/609312","pdf_url":null,"source":{"id":"https://openalex.org/S4306400312","display_name":"Virtual Community of Pathological Anatomy (University of Castilla La Mancha)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I79189158","host_organization_name":"University of Castilla-La Mancha","host_organization_lineage":["https://openalex.org/I79189158"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:air.unimi.it:2434/167367","is_oa":true,"landing_page_url":"http://hdl.handle.net/2434/167367","pdf_url":null,"source":{"id":"https://openalex.org/S4306400516","display_name":"Archivio Istituzionale della Ricerca (Universita Degli Studi Di Milano)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I189158943","host_organization_name":"University of Milan","host_organization_lineage":["https://openalex.org/I189158943"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11","score":0.5299999713897705}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":73,"referenced_works":["https://openalex.org/W11015970","https://openalex.org/W89290689","https://openalex.org/W148742480","https://openalex.org/W181581293","https://openalex.org/W649818166","https://openalex.org/W656727543","https://openalex.org/W901825327","https://openalex.org/W1517403807","https://openalex.org/W1575147392","https://openalex.org/W1594759473","https://openalex.org/W1662321012","https://openalex.org/W1753954321","https://openalex.org/W1808551600","https://openalex.org/W1967203031","https://openalex.org/W1968535318","https://openalex.org/W1970311684","https://openalex.org/W1972892940","https://openalex.org/W1981729214","https://openalex.org/W1982528487","https://openalex.org/W1991014733","https://openalex.org/W1998275129","https://openalex.org/W1998344784","https://openalex.org/W1998488130","https://openalex.org/W2003313595","https://openalex.org/W2008179227","https://openalex.org/W2009802784","https://openalex.org/W2010713031","https://openalex.org/W2012435411","https://openalex.org/W2018136562","https://openalex.org/W2023198778","https://openalex.org/W2027132750","https://openalex.org/W2027239222","https://openalex.org/W2039707252","https://openalex.org/W2041926312","https://openalex.org/W2043718735","https://openalex.org/W2046528357","https://openalex.org/W2046859243","https://openalex.org/W2050481477","https://openalex.org/W2051497316","https://openalex.org/W2055118264","https://openalex.org/W2059755416","https://openalex.org/W2066179673","https://openalex.org/W2073304254","https://openalex.org/W2076666589","https://openalex.org/W2077736526","https://openalex.org/W2077900714","https://openalex.org/W2086298431","https://openalex.org/W2087584322","https://openalex.org/W2089829222","https://openalex.org/W2095228559","https://openalex.org/W2100827192","https://openalex.org/W2105054725","https://openalex.org/W2110689112","https://openalex.org/W2112144602","https://openalex.org/W2116801612","https://openalex.org/W2118849262","https://openalex.org/W2120271468","https://openalex.org/W2137598725","https://openalex.org/W2140456006","https://openalex.org/W2146961013","https://openalex.org/W2179225387","https://openalex.org/W2315896265","https://openalex.org/W2403017595","https://openalex.org/W2504912136","https://openalex.org/W2525138856","https://openalex.org/W2592791290","https://openalex.org/W2787176649","https://openalex.org/W2897725581","https://openalex.org/W2964218936","https://openalex.org/W3105602197","https://openalex.org/W3142812456","https://openalex.org/W4285719527","https://openalex.org/W4300657400"],"related_works":["https://openalex.org/W2176389738","https://openalex.org/W2050770628","https://openalex.org/W2401850100","https://openalex.org/W4294596620","https://openalex.org/W2338743497","https://openalex.org/W4298356814","https://openalex.org/W881700581","https://openalex.org/W2953017298","https://openalex.org/W4297850705","https://openalex.org/W2980411421"],"abstract_inverted_index":{"We":[0,131],"consider":[1],"a":[2,69,104,121,127,139,159],"modified":[3],"Cahn\u2013Hiliard":[4],"equation":[5,38],"where":[6],"the":[7,10,16,23,59,80,97,111,134,150,156],"velocity":[8],"of":[9,19,37,103,158],"order":[11],"parameter":[12],"u":[13,65],"depends":[14],"on":[15,126],"past":[17,99],"history":[18,100],"\u0394\u03bc,":[20],"\u03bc":[21],"being":[22],"chemical":[24],"potential":[25],"with":[26,79],"an":[27,146],"additional":[28],"viscous":[29],"term":[30],"\u03b1u":[31],"t":[32],",":[33],"\u03b1\u22650.":[34],"This":[35],"type":[36],"has":[39,138],"been":[40],"proposed":[41],"by":[42,68],"P.":[43],"Galenko":[44],"et":[45],"al.":[46],"to":[47,89],"model":[48],"phase":[49,129],"separation":[50],"phenomena":[51],"in":[52,96,110],"special":[53],"materials":[54],"(e.g.,":[55],"glasses).":[56],"In":[57,149],"addition,":[58],"usual":[60],"no-flux":[61],"boundary":[62,72,84],"condition":[63,73],"for":[64,76],"is":[66,87,94,107],"replaced":[67],"nonlinear":[70],"dynamic":[71],"which":[74],"accounts":[75],"possible":[77],"interactions":[78],"boundary.":[81],"The":[82],"resulting":[83],"value":[85],"problem":[86],"subject":[88],"suitable":[90,128],"initial":[91],"conditions":[92],"and":[93,119],"reformulated":[95],"so-called":[98],"space.":[101,130],"Existence":[102],"variational":[105],"solution":[106],"obtained.":[108],"Then,":[109],"case":[112,151],"\u03b1>0,":[113],"we":[114,153],"can":[115],"also":[116],"prove":[117],"uniqueness":[118],"construct":[120],"strongly":[122],"continuous":[123],"semigroup":[124],"acting":[125],"show":[132],"that":[133],"corresponding":[135],"dynamical":[136],"system":[137],"(smooth)":[140],"global":[141],"attractor":[142],"as":[143,145],"well":[144],"exponential":[147],"attractor.":[148,161],"\u03b1=0,":[152],"only":[154],"establish":[155],"existence":[157],"trajectory":[160]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2}],"updated_date":"2026-04-30T06:05:26.967640","created_date":"2025-10-10T00:00:00"}