{"id":"https://openalex.org/W7221803263","doi":"https://doi.org/10.48550/arxiv.2610.17539","title":"Sharp Damping Location for Stability of a Locally Damped Wave Equation and its Galerkin Approximation","display_name":"Sharp Damping Location for Stability of a Locally Damped Wave Equation and its Galerkin Approximation","publication_year":2026,"publication_date":"2026-10-13","ids":{"openalex":"https://openalex.org/W7221803263","doi":"https://doi.org/10.48550/arxiv.2610.17539"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.17539","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.17539","pdf_url":"https://arxiv.org/pdf/2610.17539","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/2610.17539","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5158701132","display_name":"Kangsheng Liu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liu, Kangsheng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5158733936","display_name":"Lihui Lu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I76130692","display_name":"Zhejiang University","ror":"https://ror.org/00a2xv884","country_code":"CN","type":"education","lineage":["https://openalex.org/I76130692"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lu, Lihui","raw_affiliation_strings":["School of Mathematical Sciences , Zhejiang University , Hangzhou , 310058 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences , Zhejiang University , Hangzhou , 310058 , China","institution_ids":["https://openalex.org/I76130692"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5158765212","display_name":"Qiong Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I125839683","display_name":"Beijing Institute of Technology","ror":"https://ror.org/01skt4w74","country_code":"CN","type":"education","lineage":["https://openalex.org/I125839683","https://openalex.org/I890469752"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhang, Qiong","raw_affiliation_strings":["School of Mathematics and Statistics , Beijing Institute of Technology , Beijing 100081 , China;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics , Beijing Institute of Technology , Beijing 100081 , China;","institution_ids":["https://openalex.org/I125839683"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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/T11110","display_name":"Stability and Controllability of Differential Equations","score":1.0,"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"}},"topics":[{"id":"https://openalex.org/T11110","display_name":"Stability and Controllability of Differential Equations","score":1.0,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":9.999999747378752e-05,"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/T11654","display_name":"Advanced Mathematical Physics Problems","score":0.0,"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/wave-equation","display_name":"wave equation","score":0.5580000281333923},{"id":"https://openalex.org/keywords/localized-damping","display_name":"localized damping","score":0.5839999914169312},{"id":"https://openalex.org/keywords/energy-decay-rate","display_name":"energy decay rate","score":0.7639999985694885},{"id":"https://openalex.org/keywords/exponential-stability","display_name":"exponential stability","score":0.8320000171661377},{"id":"https://openalex.org/keywords/semigroups","display_name":"semigroups","score":0.6850000023841858},{"id":"https://openalex.org/keywords/galerkin-method","display_name":"Galerkin method","score":0.8259999752044678},{"id":"https://openalex.org/keywords/variable-coefficient-wave-equation","display_name":"variable coefficient wave equation","score":0.6800000071525574},{"id":"https://openalex.org/keywords/dirichlet-boundary-conditions","display_name":"Dirichlet boundary conditions","score":0.800000011920929},{"id":"https://openalex.org/keywords/resolvent-estimates","display_name":"resolvent estimates","score":0.8809999823570251},{"id":"https://openalex.org/keywords/harmonic-oscillator","display_name":"harmonic oscillator","score":0.7979999780654907},{"id":"https://openalex.org/keywords/uniform-stability","display_name":"uniform stability","score":0.3970000147819519}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7422999739646912},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.7217000126838684},{"id":"https://openalex.org/C207405024","wikidata":"https://www.wikidata.org/wiki/Q207348","display_name":"Semigroup","level":2,"score":0.6370000243186951},{"id":"https://openalex.org/C128803854","wikidata":"https://www.wikidata.org/wiki/Q1307821","display_name":"Eigenfunction","level":3,"score":0.6187000274658203},{"id":"https://openalex.org/C66908091","wikidata":"https://www.wikidata.org/wiki/Q1127660","display_name":"Damped wave","level":3,"score":0.5655999779701233},{"id":"https://openalex.org/C196267783","wikidata":"https://www.wikidata.org/wiki/Q4392490","display_name":"Resolvent","level":2,"score":0.5605999827384949},{"id":"https://openalex.org/C186899397","wikidata":"https://www.wikidata.org/wiki/Q1491980","display_name":"Galerkin method","level":3,"score":0.524399995803833},{"id":"https://openalex.org/C59696629","wikidata":"https://www.wikidata.org/wiki/Q193846","display_name":"Wave equation","level":2,"score":0.5048999786376953},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.4884999990463257},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4629000127315521},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.4230000078678131},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.4196999967098236},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.40310001373291016},{"id":"https://openalex.org/C110342517","wikidata":"https://www.wikidata.org/wiki/Q1129902","display_name":"Constant coefficients","level":2,"score":0.39879998564720154},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.3939000070095062},{"id":"https://openalex.org/C2781302577","wikidata":"https://www.wikidata.org/wiki/Q209","display_name":"Rectangle","level":2,"score":0.3659000098705292},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.365200012922287},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.35929998755455017},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.3203999996185303},{"id":"https://openalex.org/C167964875","wikidata":"https://www.wikidata.org/wiki/Q17011487","display_name":"Exponential stability","level":3,"score":0.32010000944137573},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.3167000114917755},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.3138999938964844},{"id":"https://openalex.org/C180188523","wikidata":"https://www.wikidata.org/wiki/Q574576","display_name":"Exponential decay","level":2,"score":0.31139999628067017},{"id":"https://openalex.org/C80845027","wikidata":"https://www.wikidata.org/wiki/Q15144062","display_name":"Continuous spectrum","level":2,"score":0.3025999963283539},{"id":"https://openalex.org/C41700454","wikidata":"https://www.wikidata.org/wiki/Q1852282","display_name":"Reflection coefficient","level":2,"score":0.28209999203681946},{"id":"https://openalex.org/C55649039","wikidata":"https://www.wikidata.org/wiki/Q902618","display_name":"Floquet theory","level":3,"score":0.28200000524520874},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.2676999866962433},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.2669000029563904},{"id":"https://openalex.org/C75235859","wikidata":"https://www.wikidata.org/wiki/Q582659","display_name":"Exponential growth","level":2,"score":0.2531999945640564},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.2513999938964844}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.17539","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.17539","pdf_url":"https://arxiv.org/pdf/2610.17539","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.2610.17539","doi":"https://doi.org/10.48550/arxiv.2610.17539","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.17539","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:2610.17539","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.17539","pdf_url":"https://arxiv.org/pdf/2610.17539","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/G220452521","display_name":null,"funder_award_id":"12271035","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5014072334","display_name":null,"funder_award_id":"12171413","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7459741091","display_name":null,"funder_award_id":"12131008","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7221803263.pdf","grobid_xml":"https://content.openalex.org/works/W7221803263.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W2273159482","https://openalex.org/W2963257407","https://openalex.org/W2963722550","https://openalex.org/W2057077570","https://openalex.org/W1491431295","https://openalex.org/W2146477284","https://openalex.org/W2373486345","https://openalex.org/W2998089103","https://openalex.org/W2007679297","https://openalex.org/W1546601687"],"abstract_inverted_index":{"We":[0,31,49,119],"study":[1],"the":[2,5,13,21,37,40,44,52,59,62,66,72,79,83,92,96,113,122,128,132,140,146,150],"stability":[3],"of":[4,39,46,61,74,104,131,142],"variable":[6],"coefficient":[7,134],"wave":[8],"equation":[9],"$u_{tt}-u_{xx}-(1+x^{2})u_{yy}+\u03c7_{\\mathbb":[10],"G}u_{t}=0$":[11],"on":[12,25,127],"rectangle":[14],"$(-1,1)\\times(0,\u03c0)$":[15],"with":[16],"Dirichlet":[17,129],"boundary":[18],"conditions,":[19],"where":[20],"viscous":[22],"damping":[23,41,63,84],"acts":[24],"a":[26,33,109],"vertical":[27],"strip":[28,64],"$\\mathbb":[29],"G=(\u03b7-\u03b4,\u03b7+\u03b4)\\times(0,\u03c0)$.":[30],"establish":[32],"precise":[34],"relationship":[35],"between":[36],"location":[38],"region":[42,85],"and":[43,116,136],"rate":[45,98],"energy":[47],"decay.":[48],"prove":[50],"that":[51,76],"associated":[53],"semigroup":[54,93],"is":[55,86],"exponentially":[56,158],"stable":[57],"whenever":[58],"closure":[60],"meets":[65],"line":[67],"$x=0$,":[68,91],"i.e.,":[69],"$|\u03b7|\\leq\u03b4$,":[70],"despite":[71],"existence":[73],"rays":[75],"never":[77],"enter":[78],"damped":[80],"region.":[81],"When":[82],"strictly":[87],"bounded":[88],"away":[89],"from":[90,145],"decays":[94],"at":[95],"sharp":[97],"$(\\log(2+t))^{-1}$.":[99],"The":[100],"proofs":[101],"combine":[102],"separation":[103],"variables,":[105],"one-dimensional":[106],"resolvent":[107],"estimates,":[108],"reflection":[110],"identity":[111],"for":[112],"harmonic":[114],"oscillator,":[115],"Gaussian":[117],"quasimodes.":[118],"then":[120],"consider":[121],"spectral":[123],"Galerkin":[124],"approximation":[125,147],"based":[126],"eigenfunctions":[130],"constant":[133],"Laplacian":[135],"show":[137],"that,":[138],"although":[139],"sequence":[141],"semigroups":[143],"resulting":[144],"converges":[148],"to":[149,155],"continuous":[151],"one,":[152],"it":[153],"fails":[154],"be":[156],"uniformly":[157],"stable.":[159]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-15T00:00:00"}
