{"id":"https://openalex.org/W2015171991","doi":"https://doi.org/10.1137/s0036141099351334","title":"Scaling Variables and Stability of Hyperbolic Fronts","display_name":"Scaling Variables and Stability of Hyperbolic Fronts","publication_year":2000,"publication_date":"2000-01-01","ids":{"openalex":"https://openalex.org/W2015171991","doi":"https://doi.org/10.1137/s0036141099351334","mag":"2015171991"},"language":"en","primary_location":{"id":"doi:10.1137/s0036141099351334","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036141099351334","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/A5065688870","display_name":"Thierry Gallay","orcid":"https://orcid.org/0000-0001-8181-9883"},"institutions":[{"id":"https://openalex.org/I102197404","display_name":"Universit\u00e9 Paris-Sud","ror":"https://ror.org/028rypz17","country_code":"FR","type":"education","lineage":["https://openalex.org/I102197404"]},{"id":"https://openalex.org/I204730241","display_name":"Universit\u00e9 Paris Cit\u00e9","ror":"https://ror.org/05f82e368","country_code":"FR","type":"education","lineage":["https://openalex.org/I204730241"]},{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1294671590"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Thierry Gallay","raw_affiliation_strings":["CNRS et Universit\u00e9 de Paris-Sud URA 760, Analyse Num\u00e9rique et EDP B\u00e2timent 425 F-91405 Orsay Cedex, France"],"affiliations":[{"raw_affiliation_string":"CNRS et Universit\u00e9 de Paris-Sud URA 760, Analyse Num\u00e9rique et EDP B\u00e2timent 425 F-91405 Orsay Cedex, France","institution_ids":["https://openalex.org/I204730241","https://openalex.org/I102197404","https://openalex.org/I1294671590"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111667718","display_name":"Genevi\u00e8ve Raugel","orcid":null},"institutions":[{"id":"https://openalex.org/I1294671590","display_name":"Centre National de la Recherche Scientifique","ror":"https://ror.org/02feahw73","country_code":"FR","type":"funder","lineage":["https://openalex.org/I1294671590"]},{"id":"https://openalex.org/I102197404","display_name":"Universit\u00e9 Paris-Sud","ror":"https://ror.org/028rypz17","country_code":"FR","type":"education","lineage":["https://openalex.org/I102197404"]},{"id":"https://openalex.org/I204730241","display_name":"Universit\u00e9 Paris Cit\u00e9","ror":"https://ror.org/05f82e368","country_code":"FR","type":"education","lineage":["https://openalex.org/I204730241"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Genevi\u00e8ve Raugel","raw_affiliation_strings":["CNRS et Universit\u00e9 de Paris-Sud URA 760, Analyse Num\u00e9rique et EDP B\u00e2timent 425 F-91405 Orsay Cedex, France"],"affiliations":[{"raw_affiliation_string":"CNRS et Universit\u00e9 de Paris-Sud URA 760, Analyse Num\u00e9rique et EDP B\u00e2timent 425 F-91405 Orsay Cedex, France","institution_ids":["https://openalex.org/I204730241","https://openalex.org/I102197404","https://openalex.org/I1294671590"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5065688870"],"corresponding_institution_ids":["https://openalex.org/I102197404","https://openalex.org/I1294671590","https://openalex.org/I204730241"],"apc_list":null,"apc_paid":null,"fwci":3.5335,"has_fulltext":false,"cited_by_count":34,"citation_normalized_percentile":{"value":0.92536272,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"32","issue":"1","first_page":"1","last_page":"29"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.9988999962806702,"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":0.9988999962806702,"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/T11110","display_name":"Stability and Controllability of Differential Equations","score":0.9976000189781189,"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"}},{"id":"https://openalex.org/T10482","display_name":"Mathematical and Theoretical Epidemiology and Ecology Models","score":0.9783999919891357,"subfield":{"id":"https://openalex.org/subfields/2739","display_name":"Public Health, Environmental and Occupational Health"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7992457151412964},{"id":"https://openalex.org/keywords/sobolev-space","display_name":"Sobolev space","score":0.7149099707603455},{"id":"https://openalex.org/keywords/scaling","display_name":"Scaling","score":0.6682896018028259},{"id":"https://openalex.org/keywords/monotone-polygon","display_name":"Monotone polygon","score":0.6257416605949402},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5779551267623901},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.547041118144989},{"id":"https://openalex.org/keywords/interval","display_name":"Interval (graph theory)","score":0.49216359853744507},{"id":"https://openalex.org/keywords/zero","display_name":"Zero (linguistics)","score":0.4909096360206604},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.47819164395332336},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.4453585147857666},{"id":"https://openalex.org/keywords/wave-equation","display_name":"Wave equation","score":0.43794792890548706},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.4228243827819824},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3609752953052521},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.19759878516197205},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08069482445716858}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7992457151412964},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.7149099707603455},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.6682896018028259},{"id":"https://openalex.org/C2834757","wikidata":"https://www.wikidata.org/wiki/Q4925424","display_name":"Monotone polygon","level":2,"score":0.6257416605949402},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5779551267623901},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.547041118144989},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.49216359853744507},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.4909096360206604},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.47819164395332336},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.4453585147857666},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.43794792890548706},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.4228243827819824},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3609752953052521},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.19759878516197205},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08069482445716858},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/s0036141099351334","is_oa":false,"landing_page_url":"https://doi.org/10.1137/s0036141099351334","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.139.9618","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.139.9618","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ma.utexas.edu/mp_arc/c/98/98-756.ps.gz","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.339.3188","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.339.3188","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/patt-sol/9812007v1.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W61301618","https://openalex.org/W200262045","https://openalex.org/W1527296550","https://openalex.org/W1798112682","https://openalex.org/W1966988202","https://openalex.org/W1982920065","https://openalex.org/W1999891320","https://openalex.org/W2012058749","https://openalex.org/W2014863630","https://openalex.org/W2018852942","https://openalex.org/W2021474405","https://openalex.org/W2023389177","https://openalex.org/W2041451025","https://openalex.org/W2048627180","https://openalex.org/W2052172730","https://openalex.org/W2065971560","https://openalex.org/W2071706553","https://openalex.org/W2072090554","https://openalex.org/W2074505548","https://openalex.org/W2078535099","https://openalex.org/W2079949023","https://openalex.org/W2087757234","https://openalex.org/W2091636379","https://openalex.org/W2092574869","https://openalex.org/W2093201093","https://openalex.org/W2094495898","https://openalex.org/W2157813363","https://openalex.org/W2170031024","https://openalex.org/W2497646188"],"related_works":["https://openalex.org/W1984138193","https://openalex.org/W2058060618","https://openalex.org/W2315649721","https://openalex.org/W2098586728","https://openalex.org/W3114525629","https://openalex.org/W1534977344","https://openalex.org/W2363478026","https://openalex.org/W2047046892","https://openalex.org/W2951529113","https://openalex.org/W4285676930"],"abstract_inverted_index":{"We":[0,37],"consider":[1],"the":[2,51,70,87,90,93,100,120,144,158],"damped":[3],"hyperbolic":[4],"equation":[5,160],"$$":[6,27],"\\epsilon":[7],"u_{tt}":[8],"+":[9,13],"u_t":[10],"\\,=\\,":[11],"u_{xx}":[12],"\\FF(u)":[14],",":[15,20,25],"\\quad":[16,21],"x":[17],"\\in":[18],"\\mathbf{R}":[19],"t":[22],"\\ge":[23,83],"0":[24],"\\leqno(1)":[26],"where":[28],"$\\epsilon$":[29],"is":[30,48,77,103,138],"a":[31,59,80,111,133],"positive,":[32],"not":[33],"necessarily":[34],"small":[35],"parameter.":[36],"assume":[38],"that":[39,46,99,119],"$\\FF(0)":[40],"=":[41,43,165],"\\FF(1)":[42],"0$":[44,166],"and":[45,73,131,143],"$\\FF$":[47],"concave":[49],"on":[50,175],"interval":[52],"[0,1].":[53],"Under":[54],"these":[55],"hypotheses,":[56],"(1)":[57,180],"has":[58],"family":[60,76],"of":[61,89,140,157,171],"monotone":[62],"traveling":[63,101],"wave":[64,102],"solutions":[65,150],"(or":[66],"propagating":[67],"fronts)":[68],"connecting":[69],"equilibria":[71],"u=0":[72],"u=1.":[74],"This":[75],"indexed":[78],"by":[79,162],"parameter":[81],"$c":[82],"c_*$":[84],"related":[85],"to":[86,108,123],"speed":[88],"front.":[91],"In":[92,115,147],"critical":[94],"case":[95],"$c=c_*$,":[96],"we":[97,117],"prove":[98],"asymptotically":[104],"stable":[105],"with":[106],"respect":[107],"perturbations":[109,121],"in":[110,167,182],"weighted":[112],"Sobolev":[113],"space.":[114],"addition,":[116],"show":[118],"decay":[122],"zero":[124],"like":[125,155],"$t^{-3/2}$":[126],"as":[127],"$t":[128],"\\to":[129],"+\\infty$":[130],"approach":[132],"universal":[134],"self-similar":[135,183],"profile,":[136],"which":[137],"independent":[139],"$\\epsilon$,":[141],"F,":[142],"initial":[145],"data.":[146],"particular,":[148],"our":[149,172],"behave":[151],"for":[152,179],"large":[153],"times":[154],"those":[156],"parabolic":[159],"obtained":[161],"setting":[163],"$\\epsilon":[164],"(1).":[168],"The":[169],"proof":[170],"results":[173],"relies":[174],"various":[176],"energy":[177],"estimates":[178],"rewritten":[181],"variables":[184],"$x/\\sqrt{t}$,":[185],"$\\log":[186],"t$.":[187]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}