{"id":"https://openalex.org/W1994909631","doi":"https://doi.org/10.1137/s003613990241552x","title":"Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations","display_name":"Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations","publication_year":2004,"publication_date":"2004-01-01","ids":{"openalex":"https://openalex.org/W1994909631","doi":"https://doi.org/10.1137/s003613990241552x","mag":"1994909631"},"language":"en","primary_location":{"id":"doi:10.1137/s003613990241552x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s003613990241552x","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Applied Mathematics","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/A5021625363","display_name":"Chris Budd","orcid":"https://orcid.org/0000-0003-4536-1662"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"C. J. Budd","raw_affiliation_strings":["Department of Mathematical Sciences"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5009177807","display_name":"Jim Williams","orcid":"https://orcid.org/0000-0002-1464-2317"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"J. F. Williams","raw_affiliation_strings":["Department of Mathematical Sciences"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5043327432","display_name":"Victor A. Galaktionov","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"V. A. Galaktionov","raw_affiliation_strings":["Department of Mathematical Sciences"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.7618,"has_fulltext":false,"cited_by_count":67,"citation_normalized_percentile":{"value":0.88848771,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"64","issue":"5","first_page":"1775","last_page":"1809"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9991999864578247,"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"}},"topics":[{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9991999864578247,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9947999715805054,"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/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.988099992275238,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6992532014846802},{"id":"https://openalex.org/keywords/parabolic-partial-differential-equation","display_name":"Parabolic partial differential equation","score":0.6166152954101562},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5959696173667908},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5557389855384827},{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.5014777183532715},{"id":"https://openalex.org/keywords/initial-value-problem","display_name":"Initial value problem","score":0.49384361505508423},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.48496460914611816},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.4636840224266052},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.4542733132839203},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.43178120255470276},{"id":"https://openalex.org/keywords/heat-equation","display_name":"Heat equation","score":0.4273775815963745},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.40086960792541504},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3489139676094055},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.2594485580921173},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2033558189868927}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6992532014846802},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.6166152954101562},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5959696173667908},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5557389855384827},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.5014777183532715},{"id":"https://openalex.org/C26955809","wikidata":"https://www.wikidata.org/wiki/Q10167591","display_name":"Initial value problem","level":2,"score":0.49384361505508423},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.48496460914611816},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.4636840224266052},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.4542733132839203},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.43178120255470276},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.4273775815963745},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.40086960792541504},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3489139676094055},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.2594485580921173},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2033558189868927},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"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":3,"locations":[{"id":"doi:10.1137/s003613990241552x","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s003613990241552x","pdf_url":null,"source":{"id":"https://openalex.org/S77200987","display_name":"SIAM Journal on Applied Mathematics","issn_l":"0036-1399","issn":["0036-1399","1095-712X"],"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 Applied Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.57.7392","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.7392","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ftp://ftp.maths.bath.ac.uk/pub/preprints/maths0210.ps.gz","raw_type":"text"},{"id":"pmh:oai:bibcyt.ucla.edu.ve:T070300013564/13","is_oa":false,"landing_page_url":"http://bibcyt.ucla.edu.ve/cgi-win/be_alex.exe?Acceso=T070300013564/13&Nombrebd=Bciucla","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":47,"referenced_works":["https://openalex.org/W88773199","https://openalex.org/W91074586","https://openalex.org/W122089080","https://openalex.org/W183319324","https://openalex.org/W240029891","https://openalex.org/W569093513","https://openalex.org/W1482890868","https://openalex.org/W1540969907","https://openalex.org/W1564725221","https://openalex.org/W1565688698","https://openalex.org/W1913455435","https://openalex.org/W1982544175","https://openalex.org/W1991252806","https://openalex.org/W1999536242","https://openalex.org/W2009060935","https://openalex.org/W2010142289","https://openalex.org/W2012270170","https://openalex.org/W2013921840","https://openalex.org/W2017353585","https://openalex.org/W2023459929","https://openalex.org/W2034914590","https://openalex.org/W2044580900","https://openalex.org/W2047189062","https://openalex.org/W2047296116","https://openalex.org/W2049966323","https://openalex.org/W2057078612","https://openalex.org/W2061507957","https://openalex.org/W2062162218","https://openalex.org/W2067265108","https://openalex.org/W2088302643","https://openalex.org/W2093146030","https://openalex.org/W2094820190","https://openalex.org/W2112390231","https://openalex.org/W2118678892","https://openalex.org/W2118818801","https://openalex.org/W2147500254","https://openalex.org/W2148899926","https://openalex.org/W2160964994","https://openalex.org/W2798910916","https://openalex.org/W3187596537","https://openalex.org/W4205717468","https://openalex.org/W4210992751","https://openalex.org/W4243412425","https://openalex.org/W4246915629","https://openalex.org/W4249641605","https://openalex.org/W4252339980","https://openalex.org/W4302061159"],"related_works":["https://openalex.org/W2985697654","https://openalex.org/W1548680320","https://openalex.org/W2026330468","https://openalex.org/W2970108133","https://openalex.org/W2895777763","https://openalex.org/W4310656211","https://openalex.org/W2158167088","https://openalex.org/W2122308748","https://openalex.org/W1511736731","https://openalex.org/W2090283541"],"abstract_inverted_index":{"We":[0,69,163],"study":[1],"the":[2,17,62,75,78,97,121,124,134,141,156,179,185,194,199,223],"Cauchy":[3],"problem":[4],"in":[5,55,64,72,99,120],"$\\re":[6],"\\times":[7],"\\re_+$":[8],"for":[9,133,187,201],"one-dimensional":[10],"2mth-order,":[11],"m>1,":[12],"semilinear":[13,152],"parabolic":[14,81],"PDEs":[15,200],"of":[16,77,123,184,198,240,254],"form":[18],"($D_x=\\partial/\\partial":[19],"x$)":[20],"\\[":[21],"u_t":[22,38],"=":[23,39,84,90,117,129],"\\Dx":[24,40],"u":[25,41],"+":[26,42,86,92,131],"|u|^{p-1}u,":[27],"\\,\\,\\,\\mbox{where}":[28],"\\,\\,":[29],"\\,\\,\\,p":[30],">":[31],"1,":[32],"\\quad":[33,37],"\\mbox{":[34],"and":[35,88,127,189,193,249],"}":[36],"e^u":[43],"\\]":[44],"with":[45,251],"bounded":[46],"initial":[47],"data":[48],"u0":[49],"(x).":[50],"Specifically,":[51],"we":[52],"are":[53,167,203],"interested":[54],"those":[56,218],"solutions":[57,76,177,197,253],"that":[58,154,165,217],"blow":[59],"up":[60,87],"at":[61,168],"origin":[63],"a":[65,208,228,255],"finite":[66],"time":[67,195],"T.":[68],"show":[70,164],"that,":[71],"contrast":[73],"to":[74,178,232],"classical":[79],"second-order":[80],"equations":[82],"ut":[83,89],"uxx":[85,91],"eu":[93],"from":[94],"combustion":[95],"theory,":[96],"blow-up":[98,113,242],"their":[100],"higher-order":[101,143,159],"counterparts":[102],"is":[103,140,215],"asymptotically":[104],"self-similar.":[105],"In":[106],"particular,":[107],"there":[108,166],"exist":[109],"exact":[110],"nontrivial":[111,175],"self-similar":[112,176,234],"solutions,":[114],"u*":[115],"(x,t)":[116],"(T-t)-1/(p-1)f":[118],"(y)":[119],"case":[122],"polynomial":[125],"nonlinearity":[126],"u(x,t)":[128],"-ln(T-t)":[130],"f(y)":[132,149],"exponential":[135],"nonlinearity,":[136],"where":[137],"y=":[138],"x/(T-t)1/2m":[139],"backward":[142],"heat":[144],"kernel":[145],"variable.":[146],"The":[147],"profiles":[148],"satisfy":[150],"related":[151],"ODEs":[153,186],"share":[155],"same":[157],"non--self-adjoint":[158],"linear":[160],"differential":[161],"operators.":[162],"least":[169],"$2":[170],"\\lfloor":[171],"\\frac":[172],"m2":[173],"\\rfloor$":[174],"full":[180],"PDEs.":[181],"Numerical":[182],"solution":[183],"m=2":[188,202],"3":[190],"supports":[191],"this,":[192],"dependent":[196],"then":[204],"studied":[205],"by":[206,247],"using":[207],"scale":[209],"invariant":[210],"adaptive":[211],"numerical":[212],"method.":[213],"It":[214],"shown":[216],"functions":[219],"f(y),":[220],"which":[221],"have":[222],"simplest":[224],"spatial":[225],"shape":[226],"(e.g.,":[227],"single":[229],"maximum),":[230],"correspond":[231],"stable":[233],"solutions.":[235],"A":[236],"further":[237],"countable":[238],"subset":[239],"nonsimilarity":[241],"patterns":[243],"can":[244],"be":[245],"constructed":[246],"linearization":[248],"matching":[250],"similarity":[252],"first-order":[256],"Hamilton--Jacobi":[257],"equation.":[258]},"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":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":5}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}