{"id":"https://openalex.org/W7204593571","doi":"https://doi.org/10.48550/arxiv.2608.26657","title":"Heat equations in spectral Barron spaces","display_name":"Heat equations in spectral Barron spaces","publication_year":2026,"publication_date":"2026-08-27","ids":{"openalex":"https://openalex.org/W7204593571","doi":"https://doi.org/10.48550/arxiv.2608.26657"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.26657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26657","pdf_url":"https://arxiv.org/pdf/2608.26657","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.26657","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5023821627","display_name":"Mourad Choulli","orcid":"https://orcid.org/0000-0001-7427-0212","observed_orcids":["https://orcid.org/0000-0001-7427-0212"]},"institutions":[{"id":"https://openalex.org/I163770644","display_name":"Okayama University","ror":"https://ror.org/02pc6pc55","country_code":"JP","type":"education","lineage":["https://openalex.org/I163770644"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Choulli, Mourad","raw_affiliation_strings":["Department of Mathematics , Okayama University , 3-1-1 Tsushima-naka , Kita-ku , Okayama 700-8530 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Okayama University , 3-1-1 Tsushima-naka , Kita-ku , Okayama 700-8530 , Japan","institution_ids":["https://openalex.org/I163770644"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5148513920","display_name":"Shuai Lu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I163770644","display_name":"Okayama University","ror":"https://ror.org/02pc6pc55","country_code":"JP","type":"education","lineage":["https://openalex.org/I163770644"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Lu, Shuai","raw_affiliation_strings":["Department of Mathematics , Okayama University , 3-1-1 Tsushima-naka , Kita-ku , Okayama 700-8530 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Okayama University , 3-1-1 Tsushima-naka , Kita-ku , Okayama 700-8530 , Japan","institution_ids":["https://openalex.org/I163770644"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5123653501","display_name":"Hiroshi Takase","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I163770644","display_name":"Okayama University","ror":"https://ror.org/02pc6pc55","country_code":"JP","type":"education","lineage":["https://openalex.org/I163770644"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Takase, Hiroshi","raw_affiliation_strings":["Department of Mathematics , Okayama University , 3-1-1 Tsushima-naka , Kita-ku , Okayama 700-8530 , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Okayama University , 3-1-1 Tsushima-naka , Kita-ku , Okayama 700-8530 , Japan","institution_ids":["https://openalex.org/I163770644"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I163770644"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12589","display_name":"Differential Equations and Boundary Problems","score":0.8550000190734863,"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/T12589","display_name":"Differential Equations and Boundary Problems","score":0.8550000190734863,"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.030500000342726707,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.02759999968111515,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/heat-equation","display_name":"heat equation","score":0.7170000076293945},{"id":"https://openalex.org/keywords/well-posedness","display_name":"well-posedness","score":0.9549999833106995},{"id":"https://openalex.org/keywords/existence-and-uniqueness-of-solutions","display_name":"existence and uniqueness of solutions","score":0.7670000195503235},{"id":"https://openalex.org/keywords/stability-estimates","display_name":"stability estimates","score":0.718999981880188},{"id":"https://openalex.org/keywords/backward-heat-equation","display_name":"backward heat equation","score":0.9760000109672546},{"id":"https://openalex.org/keywords/inverse-parabolic-problem","display_name":"parabolic inverse problem","score":0.949999988079071},{"id":"https://openalex.org/keywords/time-fractional-heat-equation","display_name":"time-fractional heat equation","score":0.8489999771118164},{"id":"https://openalex.org/keywords/anomalous-diffusion","display_name":"anomalous diffusion","score":0.8870000243186951},{"id":"https://openalex.org/keywords/heat-kernel","display_name":"heat kernel","score":0.7670000195503235},{"id":"https://openalex.org/keywords/decay-estimates","display_name":"decay estimates","score":0.8730000257492065},{"id":"https://openalex.org/keywords/shallow-neural-networks","display_name":"shallow neural networks","score":0.796999990940094},{"id":"https://openalex.org/keywords/elliptic-regularity","display_name":"elliptic regularity","score":0.8029999732971191}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7215999960899353},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.6904000043869019},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6021999716758728},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.5368000268936157},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.5004000067710876},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.46720001101493835},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.44920000433921814},{"id":"https://openalex.org/C142730499","wikidata":"https://www.wikidata.org/wiki/Q934367","display_name":"Function space","level":2,"score":0.3971000015735626},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.39070001244544983},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.390500009059906},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.3799000084400177},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3736000061035156},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.37310001254081726},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.3675999939441681},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.3619999885559082},{"id":"https://openalex.org/C183212220","wikidata":"https://www.wikidata.org/wiki/Q3345669","display_name":"Heat kernel","level":2,"score":0.3598000109195709},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.3425000011920929},{"id":"https://openalex.org/C98306654","wikidata":"https://www.wikidata.org/wiki/Q2409122","display_name":"Spectral theory","level":3,"score":0.3407000005245209},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.31049999594688416},{"id":"https://openalex.org/C2164484","wikidata":"https://www.wikidata.org/wiki/Q5170150","display_name":"Core (optical fiber)","level":2,"score":0.29809999465942383},{"id":"https://openalex.org/C41949839","wikidata":"https://www.wikidata.org/wiki/Q1756677","display_name":"Stability theory","level":3,"score":0.2761000096797943},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.26350000500679016},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.26339998841285706},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.2596000134944916},{"id":"https://openalex.org/C23463724","wikidata":"https://www.wikidata.org/wiki/Q2308831","display_name":"Spectral method","level":2,"score":0.25119999051094055},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.25099998712539673},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.25040000677108765}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.26657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26657","pdf_url":"https://arxiv.org/pdf/2608.26657","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2608.26657","doi":"https://doi.org/10.48550/arxiv.2608.26657","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.26657","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.26657","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.26657","pdf_url":"https://arxiv.org/pdf/2608.26657","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G208158153","display_name":null,"funder_award_id":"92570106","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2895446663","display_name":null,"funder_award_id":"JP23KK0049","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G7742448768","display_name":null,"funder_award_id":"23JC1400501","funder_id":"https://openalex.org/F4320321885","funder_display_name":"Science and Technology Commission of Shanghai Municipality"},{"id":"https://openalex.org/G8473573580","display_name":"\u66f3\u8239\u306b\u304a\u3051\u308b\u5404\u7a2e\u63a8\u9032\u64cd\u8235\u65b9\u5f0f\u306e\u6bd4\u8f03\u7814\u7a76","funder_award_id":"54052","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"},{"id":"https://openalex.org/G998101322","display_name":"Exploration of a new conditional stability theory and its application to inverse problems","funder_award_id":"25K17280","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320310925","display_name":"Universit\u00e9 de Lorraine","ror":"https://ror.org/04vfs2w97"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321851","display_name":"Fudan University","ror":"https://ror.org/013q1eq08"},{"id":"https://openalex.org/F4320321885","display_name":"Science and Technology Commission of Shanghai Municipality","ror":"https://ror.org/03kt66j61"},{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204593571.pdf","grobid_xml":"https://content.openalex.org/works/W7204593571.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4416103258"],"related_works":["https://openalex.org/W4321605975","https://openalex.org/W4221140970","https://openalex.org/W1996450365","https://openalex.org/W2081863784","https://openalex.org/W4403482202","https://openalex.org/W7171931598","https://openalex.org/W3022247095","https://openalex.org/W4301489764","https://openalex.org/W4406211015","https://openalex.org/W2811461037"],"abstract_inverted_index":{"Spectral":[0],"Barron":[1,80,150,209],"spaces,":[2],"characterized":[3],"by":[4],"an":[5,38],"\\(L^1\\)-based":[6],"Fourier-Lebesgue":[7],"norm,":[8],"have":[9,35],"earned":[10],"significant":[11],"attention":[12],"in":[13],"approximation":[14],"theory":[15,50],"due":[16],"to":[17,21,159],"their":[18],"remarkable":[19],"capacity":[20],"represent":[22],"functions":[23],"via":[24],"shallow":[25],"neural":[26],"networks":[27],"with":[28],"controlled":[29],"complexity.":[30],"Meanwhile,":[31],"recent":[32],"theoretical":[33,201],"advances":[34],"firmly":[36],"established":[37],"intrinsic":[39],"and":[40,47,67,96,107,174],"profound":[41],"connection":[42],"between":[43],"these":[44],"function":[45],"spaces":[46,81,210],"the":[48,61,71,78,87,93,104,119,133,140,148,161,189,200,207],"regularity":[49,196],"of":[51,73,90,98,135,165,188,203],"elliptic":[52],"partial":[53],"differential":[54],"equations.":[55],"Building":[56],"upon":[57],"this":[58,138,181],"foundational":[59],"interplay,":[60],"present":[62],"work":[63],"undertakes":[64],"a":[65,113,127,186],"systematic":[66],"comprehensive":[68],"investigation":[69],"into":[70],"well-posedness":[72],"heat":[74,121,167,191],"equations":[75],"formulated":[76],"within":[77,147,206],"spectral":[79,149,208],"framework.":[82],"Specifically,":[83],"we":[84,125,154,184],"rigorously":[85],"establish":[86],"core":[88],"aspects":[89],"well-posedness,":[91],"including":[92],"existence,":[94],"uniqueness,":[95],"stability":[97,130,142],"solutions,":[99],"under":[100],"suitable":[101],"assumptions":[102],"on":[103],"source":[105],"terms":[106],"conductivity":[108],"coefficients.":[109],"We":[110],"also":[111],"investigate":[112],"typical":[114],"parabolic":[115],"inverse":[116,145],"problem,":[117],"namely":[118],"backward":[120],"equation,":[122],"for":[123,144],"which":[124,169],"derive":[126],"logarithmic":[128],"conditional":[129],"estimate.":[131],"To":[132],"best":[134],"our":[136,156],"knowledge,":[137],"constitutes":[139],"first":[141],"estimate":[143],"problems":[146],"space":[151],"setting.":[152,211],"Moreover,":[153],"extend":[155],"analytical":[157],"results":[158],"address":[160],"more":[162],"intricate":[163],"setting":[164],"time-fractional":[166],"equations,":[168],"govern":[170],"anomalous":[171],"diffusion":[172],"phenomena":[173],"introduce":[175],"nonlocal":[176],"temporal":[177],"memory":[178],"effects.":[179],"In":[180],"extended":[182],"context,":[183],"provide":[185],"characterization":[187],"corresponding":[190],"kernels,":[192],"deriving":[193],"decay":[194],"estimates,":[195],"properties,":[197],"thereby":[198],"enriching":[199],"landscape":[202],"evolutionary":[204],"PDEs":[205]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-29T00:00:00"}