{"id":"https://openalex.org/W3007000234","doi":"https://doi.org/10.1137/19m1253654","title":"Blowup for a Fourth-Order Parabolic Equation with Gradient Nonlinearity","display_name":"Blowup for a Fourth-Order Parabolic Equation with Gradient Nonlinearity","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3007000234","doi":"https://doi.org/10.1137/19m1253654","mag":"3007000234"},"language":"en","primary_location":{"id":"doi:10.1137/19m1253654","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1253654","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058663855","display_name":"Kazuhiro Ishige","orcid":"https://orcid.org/0000-0003-4799-7592"},"institutions":[{"id":"https://openalex.org/I74801974","display_name":"The University of Tokyo","ror":"https://ror.org/057zh3y96","country_code":"JP","type":"education","lineage":["https://openalex.org/I74801974"]}],"countries":["JP"],"is_corresponding":true,"raw_author_name":"Kazuhiro Ishige","raw_affiliation_strings":["The University of Tokyo,"],"affiliations":[{"raw_affiliation_string":"The University of Tokyo,","institution_ids":["https://openalex.org/I74801974"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065790102","display_name":"Nobuhito Miyake","orcid":"https://orcid.org/0000-0002-0082-5163"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nobuhito Miyake","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5056830225","display_name":"Shinya Okabe","orcid":"https://orcid.org/0000-0003-0103-7449"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shinya Okabe","raw_affiliation_strings":["SCI - Mathematics"],"affiliations":[{"raw_affiliation_string":"SCI - Mathematics","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5058663855"],"corresponding_institution_ids":["https://openalex.org/I74801974"],"apc_list":null,"apc_paid":null,"fwci":2.9238,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.91021945,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"52","issue":"1","first_page":"927","last_page":"953"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9997000098228455,"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"}},{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9959999918937683,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/nabla-symbol","display_name":"Nabla symbol","score":0.9798613786697388},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8263802528381348},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.6456090211868286},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.5947611331939697},{"id":"https://openalex.org/keywords/cauchy-problem","display_name":"Cauchy problem","score":0.5681139826774597},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5444818735122681},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5337328314781189},{"id":"https://openalex.org/keywords/parabolic-partial-differential-equation","display_name":"Parabolic partial differential equation","score":0.4958730638027191},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.35437893867492676},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.3240922689437866},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.2171030342578888},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.13502264022827148},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.05338636040687561},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.038630664348602295}],"concepts":[{"id":"https://openalex.org/C54207081","wikidata":"https://www.wikidata.org/wiki/Q937750","display_name":"Nabla symbol","level":3,"score":0.9798613786697388},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8263802528381348},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.6456090211868286},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.5947611331939697},{"id":"https://openalex.org/C153635880","wikidata":"https://www.wikidata.org/wiki/Q1412436","display_name":"Cauchy problem","level":3,"score":0.5681139826774597},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5444818735122681},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5337328314781189},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.4958730638027191},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.35437893867492676},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3240922689437866},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.2171030342578888},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.13502264022827148},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.05338636040687561},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.038630664348602295},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/19m1253654","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1253654","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematical Analysis","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G4521601337","display_name":null,"funder_award_id":"16H03916","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G5412174584","display_name":null,"funder_award_id":"JP19J10424","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G8753241139","display_name":null,"funder_award_id":"19H05599","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":14,"referenced_works":["https://openalex.org/W1848128011","https://openalex.org/W1995153412","https://openalex.org/W1995790018","https://openalex.org/W1996078453","https://openalex.org/W2009060935","https://openalex.org/W2016672830","https://openalex.org/W2020293658","https://openalex.org/W2041283199","https://openalex.org/W2059287150","https://openalex.org/W2316131184","https://openalex.org/W2317914386","https://openalex.org/W2333035580","https://openalex.org/W2525657796","https://openalex.org/W2614938920"],"related_works":["https://openalex.org/W2078366360","https://openalex.org/W2400581667","https://openalex.org/W2049761058","https://openalex.org/W4213342640","https://openalex.org/W2394174381","https://openalex.org/W4394654496","https://openalex.org/W2020390959","https://openalex.org/W2345019225","https://openalex.org/W2503576560","https://openalex.org/W2144182429"],"abstract_inverted_index":{"Let":[0],"$u$":[1,44],"be":[2,46],"a":[3,10,32,74],"solution":[4,43],"to":[5,45,83],"the":[6,36,42,68,78,84],"Cauchy":[7,85],"problem":[8],"for":[9,35],"fourth-order":[11],"nonlinear":[12],"parabolic":[13],"equation":[14],"$\\partial_t":[15],"u+(-\\Delta)^2u=-\\nabla\\cdot(|\\nabla":[16],"u|^{p-2}\\nabla":[17],"u)$":[18],"on":[19,67,77],"${\\bf":[20],"R}^N$,":[21],"where":[22],"$p>2$":[23],"and":[24,62],"$N\\ge":[25],"1$.":[26],"In":[27],"this":[28],"paper":[29],"we":[30,49,63],"give":[31,73],"sufficient":[33,75],"condition":[34,76],"maximal":[37],"existence":[38,79],"time":[39],"$T_M(u)$":[40],"of":[41,80],"finite.":[47],"Furthermore,":[48],"show":[50],"that":[51],"if":[52],"$T_M(u)<\\infty$,":[53],"then":[54],"$\\|\\nabla":[55],"u(t)\\|_{L^\\infty({\\bf":[56],"R}^N)}$":[57],"blows":[58],"up":[59],"at":[60],"$t=T_M(u)$,":[61],"obtain":[64],"lower":[65],"estimates":[66],"blow-up":[69],"rate.":[70],"We":[71],"also":[72],"global-in-time":[81],"solutions":[82],"problem.":[86]},"counts_by_year":[{"year":2026,"cited_by_count":5},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":2}],"updated_date":"2026-04-09T08:11:56.329763","created_date":"2025-10-10T00:00:00"}
